.-J   V 


University  of  California  •  Berkeley 
Gift  of 


Miss  Helen  Pardee 


T  II  E 

SCHOLAR'S  ARITHMETIC; 

O  R 

FEDERAL  ACCOUNTANT, 

CONTAINING, 

I.  Common  Arithmetic,  fHE  Rules  and  iLLvstRAfioys. 

n.  Examples  and  Answers  ivifn  rlank   spaces,  sufficient  for  fHEiR 

OPERATIONS  nr  riiE  SCHOLAR. 

III.    To  EACH  MULB  A  SvVPLKMENTy    COMPREHENDING,    L  QUESTIONS    ON 

THE  NATURE   OF  THE  RULE,    iTS  USE,  AND   THE  MANNER  OF 

ITS  OPERATIONS.     2.  EXERCISES. 

XV.  Federal  Money, ^F/^i/  rules  for  all  the  various  operations  in  it 
-^To  REDUCE  Federal  to  Old  Lawful  and  Old 

Lawful  To  Federal  Money.  . 

V.  Interest  cast  in  Yehkral  Money,  /f/t'// Compound  Multiplication, 
Compound  Division  and  Practice,  wrought  in  Old  Lawful  and 

IN  Federal  Money,  the  same  questions  being  put,  in  sepa- 

rate  columns,  on  the  same  page,  in  each  kind  of  monet,  bi" 

which  these  two  modes  of  account  become  contrasted, 

and  the  great  advantage  gained  br  reckoning 

/:/ Federal  Money  easilt  discerned. 

VI.  Demonstrations  sr  engravings  of  the  reason  and  nature  of  the 

VARIOUS  STEPS  IN  THE  EXTRACTION  OF  THE  SQUARE  AND  CuBE  RooTS, 
NOT  To  BE  FOUND  IN   ANT   OTHER    TREATISE    ON   ARITHMETIC. 

VII.  Forms  of  Notes,  Deeds,  Bonds^  and  other  instruments  of  writing, 

THE  WHOLE  IN  A  FORM  AND  METHOD  ALTOGETHER  NEW, 

FOR  THE  EASE  OF  THE  MASTER  AND  THE  GREATER 
PROGRESS  OF   THE  SCHOLAR. 

—  >  ^A  y  -J'.'r  ^  -5:i-  ■~r  — 

BY  DANIEL  ADAMS,  M.  B. 

—  ^  -;;'?  ^  i\k  .f  -Jii-  >>•  — 
FIFTH  EDITION. 

PUBLISHED  ACCORDING  TO  ACT  OF  COAGRESS. 


KEENE,  N.  H.— PRINTED  BY  JOHN  PRENTISS. 
[Proprietor  of  the  copy-riii^ht] 
EOR  HASTINGS,  KTHRIDGE  isf  BLISS, 
BOSTON. 

180». * 

1  Doli.  iinglc.J 


District  of  Massacbiiseu^  District,    to  wit  : 

^^^  TK 

U    L  s    )|  J-Je  it  remembered,   that  on   the  Nintli  day  of  September,  irt 

aL  Jl     the  26th    year   of  the   Independence  of  the  United  States  of 

lli§5=3<&l  America,  DANIEL  ADAMS,  of  the  said  District  haih  depos- 
ited in  this  Office  the  Title  of  a  book,  the  right  whereof  he  chiims  as  Author, 
in  the  following  words,  to  wit  : — "  The  Scholar's  Jiui'iiMEfic  :  or  Federal 
Accountant.  Containing,  I.  Common  Arithmetic,  the  rules  and  illustrations. 
— IL  Examples  and  Answers  with  blank  spaces  sufficient  for  their  operation 
by  the  Scholar — III.  To  each  rule  a  Supplement,  comprehending,  1.  Ques- 
tions on  the  nnture  of  the  rule,  its  use,  and  the  manner  of  its  operations — 2. 
Exercises — IV.  Federal  Money  with  rules  for  all  the  various  operations  i^i 
it,  to  reduce  Federal  to  Old  Lawful  and  Old  Lawful  to  Federal  Money  — 
V.  Interest  cast  in  Federal  Money,  with  Compound  MuUipiierition,  Com- 
pound Division,  and  Practice  wrought  in  Old  Lawful  and  in  Federal  Money, 
the  same  questions  being  put  in  separate  columns  on  the  same  page,  in  eaeh  " 
kind  of  money,  by  which  these  two  modes  of  account  become  contrasted 
and  the  great  advantage  gained  by  reckoning  in  Federal  Money  easily  dis- 
cerned— V'l.  Demonstrations  by  engravings  of  the  reason  and  nature  of 
the  various  steps  in  the  extraction  of  tlie  Square  and  Cube  Roots,  not  to  be 
found  in  any  other  treatise  on  Arithmetic. — VII.  Forms  ot  Notes,  Deeds, 
Bonds  and  other  instruments  in  writing — The  whole  in  a  form  and  method 
altogether  new,  for  the  ease  of  the  Master  and  the  greater  progress  of  tlve 
Scholar. 

By  DANIEL  ADAMS,  m.  b." 

In  conformity  to  the  act  of  the  Congress  of  the  United 
States,  entitled,  "An  act  for  the  encouragement  of  Learning,  by  securing 
the  Copies,  of  Maps,  Charts  and  Books,  to  the  Authors  and  Proprietors  of 
such  Copies,  during  the  times  therein  mentioned." 

'^.QOODAh'E.,  Clerk  of  t/ieDistrici 
A  true  Cofni  of  Record.      \  of  Massachusetts  District. 

^//c6/,N.GOODALE,  Clerk. 


RECOMMENDATIONS 


t  J- 00000  ^\>  00000  J- * 


Nciv-Salem,  Sept.  14//6,  1801. 
HAVING  attentively  examined  *'  The  Scholar's  Arithmetic,'^  I 
cheerfully  give  it  as  my  opinion,  that  it  is  well  calculated  for  the 
instruction  of  youth  ;  and  that  it  will  abridge  much  of  the  time 
now  necessary  to  be  spent  in  the  communication  and  attainment 
of  such  Arithmetical  knowledge,  as  is  proper  for  the  discharge  of 
business,  WARREN  PIERCE, 

Preceptor  of  Nenv- Salem  Acacle?ny, 

Groton  Academy,  Sept,  2,  1801. 
Sir, 
I  HAVE  perused  with  attention  **  The  Scholar'' s  Arithmetic^ 
which  you  transmitted  to  me  some  time  since.  It  is  in  my  opin- 
ion, better  calculated  to  lead  students  in  our  Schools  and  Acade- 
mies into  a  complete  knowledge  of  all  that  is  useful  in  that  branch 
of  literature,  than  any  other  work  of  the  kind,  I  have  seen. 
With  great  sincerity  I  w  ish  you  success  in  your  exertions  for  the 
promotion  of  useful  learning  ;  and  I  am  confident,  that  to  be  gcn- 
erallv  approved  your  work  needs  onlv  to  be  gencrallv  known. 

WILLIAM  M,  RICHARDSON. 
Preceptor  oj  the  Academy, 

EXTRACT 

Of  a  letter  from  the  Hon,    JOHN  IVHEELOCK,  L,  L.  D. 
President  of  Dartmouth  College,  to  the  Author. 

**  THE  Scholar's  Aridimetic  is  an  improvement  on  former 
productions  of  the  same  nature.  Its  distinctive  order  and  supple- 
ment will  help  tke  learner  in  his  progress  ;  the  pait  on  Federal 
Money  makes  it  more  useful ;  and  I  have  no  doubt  but  the  whole 
will  be  a  new  fund  of  profit  in  our  country." 

September  1th,  1807. 
THE  Scholar's  Aridimetic  contains  most  of  the  important 
Rules  of  the  Art,  and  something,  also,  of  the  curious  and  enter- 
taining kind. 


iv     .  RECOMMENDATIONS. 

The  sul)jects  are  handled  In  a  simple  and  concise  manner. 

While  the  Questions  are  few,  they  exhibit  a  considerable  va* 
riety.  While  they  are,  generally,  easy,  some  of  them  afford  scope 
for  the  exercise  of  the  Scholar's  Judgment. 

It  is  a  good  quality  of  the  Book,  that  it  has  so  much  to  do  with 
Federal  Money. 

'i'he  plan  of  showing  the  reasons  of  the  operations  in  tlie  ex- 
traction of  the  Square  and  Cube  Roots  is  good. 

DANIEL  HARDY,  Jun. 
Preceptor  of  Chesterfield  Academy, 

Extract  of  a  letter  from  the  Rev.  LAB  AN  AINSWORTH,  of 
Jaffrey,  to  the  publisher  of  the  4>th  Edition,  dated  Aug,  3,  1807. 

**  THE  Superiority  of  the  Scholar's  Arithmetic  to  any  hook 
of  the  kind  in  my  knowledge,  clearly  appears  from  its  good  ef- 
fect in  the  Schools  I  annually  visit.  Previous  to  its  introduc- 
tion. Arithmetic  was  learned  and  performed  mechanically  ;  since, 
scholars  are  able  to  give  a  rational  account  of  the  several  opera- 
tions in  Arithmetic,  which  is  the  best  proof  of  their  having  Icarn* 
ed  to  good  purpose." 


PREFACE  [TOTHE2clEDiriON]  DEDICATORY 
TO 

SCHOOL  MASTERS, 

iww  I  urn  -;  rrr  tn —    — 
GEJVTLEMEJ\r, 

After  .expressing  my  sincere  thanks  for  j^onr  kind  and  very 
ready  acceptance  of  the  first  Edition  of  the  Scholar's  Arith- 
Me  TIC,  permit  me  "now  to  offer  for  your  furdier  consideration 
and  favor,  the  Second  ELdition,  which  with  its  Corrections 
and  Additions,  it  is  hoped,  will  be  found  still  more  deserving 
of  your  approbation. 

The  testimony  of  many  respectable  Teachers  has  inspired  a 
confidence  to  believe,  that  this  work,  where  it  has  been  introduc- 
ed into  Schools,  has  proved  a  kind  assistant  towards  a  more 
speedy  and  thorough  improvement  of  Scholars  in  Numbers,  and 
at  the  same  time,  has  relieved  masters  of  a  heavy  burden  of  writ- 
ing out  Kules  and  Questions,  under  which  they  have  so  long 
laboured,  to  the  manifest  neglect  of  other  parts  of  their  Schools. 

To  answer  the  several  intentions  of  this  work,  it  will  be  neces- 
sary that  it  should  be  put  into  the  hands  of  every  Arithmetician  ; 
the  blank  after  each  example  is  designed  for  the  operation  by  the 
Scholar,  which  being  first  wrousjht  upon  a  slate,  or  waste  paper, 
he  may  afterwards  transcribe  into  his  book. 

'J'liE  Supplement  to  each  Rule  inthis  work  is  a  novelty. 
I  have  often  seen  books  with  questions  and  answers,  but  in  my 
humble  opinion,  it  is  no  evidence  that  the  Scholar  comprehends 
the  principles  of  that  science  which  is  his  study,  because  that  he 
may  be  able  to  repeat  verbatim  from  his  book  the  answer  to  a 
question  ot\  which  his  attention  has  been  exercised,  two  or  three 
hours  to  commit  to  memory.  Sttidy  is  of  but  htde  advantage 
to  the  human  mind  without  reflection.  'I'o  force  the  Scholar  in- 
to rdlcctions  of  his  own,  is  the  object  of  those  l^iestions  unan- 
swered, at  ti^c  beginning  of  each  .Supplement.  The  Exifcises 
are  designed,  tests  of  his  judgment.  'J'he  Supplements  nfcly  be 
omitted  the  first  time  going  through  the  book,  if  thought  proper, 
and  rnkcn  up  aficrwards  as  a  review. 

Thro'  t|ie  whole  it  has  been  my  greatest  care  to  make  myself 
tnteligiblc   to  the  Sciiolar ;  such  rules  and  remarks  as  Iiavc  been 


vi  PREFACE  DECICATORY. 

ompiled  from  other  authors  are  indiuled  in  quotations  ;  the 
E^camples^  many  of  them  are  extracted,  this  I  have  not  hesitated 
to  do,  when  I  found  them  suited  to  my  purpose. 

Demonstrations  of  the  reason  and  nature  of  t'le  operations 
in  the  extraction  of  the  Square  and  Cube  Roots  have  never  been 
attem])ted,  in  any  work  of  the  kind  before  to  my  knowledge.  It 
is  hoped  these  will  be  found  satisfactory. 

1  HAVE  only  to  add,  that  any  intimation  of  amcndmejits  or  de- 
fects by  the  candid  and  experienced  of  your  order,  will  be  thank- 
ililly  received  by 
Gentlemen, 

Tour  most  humble^  and 

most  obedient  scr^uant, 

DANIEL  ADAMS. 
Leominster,  (Mass.)  Oct.  1,  1802. 


FIFTH  EDITION. 

The  Fifth  Edition  is  printed  page  for  page  from  the  Fourth., 
and  the  Errors  corrected. 


k 


m 


CONTENTS. 

■"■■•  ''•'"■  ''•'"  ■••'■■  ^ "'"  "•*'*■  '*'•'"  *"'■ 

Ifntro&uction* 


NOTATION  AND  NUMERATION. 

SECTION  I. 

FUNDAMENTAL  RULES  OF  ARITHMETIC. 

PACf.. 

Shnftlc  Jddition  -----------------  ------:-'-----13 

Siibtracfio7i 1-9 

Mutti/ilication  -- -- 23 

■ — Division --..----------.---------33 

Comfiound  Addition  ----------------------------  42 

tiubtraclion  ---------------- -.--53 

SECTION  II. 

RULES    ESSENTIALLY    NECESSAIlY    FOR  EVERY  PERSON    TO    FIT  AND  QUAL- 
IFY   THEM    FOR    THE    TRANSACTION   OF    BUSINESS. 

Reduction  ---------------------------------53 

Fractions    ------------  —  .----_._   .-_.   ..,...,y^ 

Decimal  Fractions    -- ----.-----.---.  -.7  5 

Federal  Money  -------  ---  -----  ---;.--_,   .   „_  .   _-87 

Table  to  reduce  Shillings  and  Pence  to  Cents  and  Mills  --.------94, 

Tables  of  exchange  ---------------------------    -95 

Interest    ----------.-------.  ---------------gs 

Easy  Method  of  casting  interest  --------  —  ---._------   iqo 

Method  of  casting  interest  on  JVotes  and  Bonds   ivhen  partial  fiayments 

at  different  'imes  have  been  made    -------.-   .^»«--a    ]02 

Coinjiound  interest    -   -   -----   ---.-...-   -..--,.,^   ---•104 

Comfiound  Multifilicatioii    ----.-•■  —  ^ ---,..••---    io7 

Division  -   --u------------^-----.-.--   \\2 

Single  Hide  of  Three  ---«----------  —  -..-.....    jjg 

Double  Rule  of  Three  -----.------.-----«---.-   133 

Practice  --• -.,_--.-,-_-_,   -.--.,^,,^»..143 

SECTION  III. 

RULES    OCCASIONALLY    USEFUL  TO    MEN   IN  PARTICULAR  EMPLOYMENTS 

GF    LIFE. 

Involution     1 59 

Evolution , .159 

Extraction  oj  the  Sqvare  Root .  .^.    1 60 

Demonstration    of  the  Reason  and  nature  if  the  various  steps  in  M^^^fl- 

era/ion  of  extracting  the  Square  Root 161 

Extraction  of  the   Cube  Root ...:...    169 

Demonstration  oJ  the  reason  and  nature  of  the  various  str/ts  in  thr  of'.- 

cration  of  extracting  the   Cube  Root .170 

Single  Eelloivshi/i , • .   .    179 


viii  CONTENTS. 

Double  F^llovfdhlfi .181 

Barter 184 

J.o&a  and  Gain     .      .      .    ' ,187 

JJuodecimaU^or  Cross  AlultifilicaUon v  199 

Jixam/iles for  measuring    Wood 19  1 

— ■■    ■  ■ "       — Boards 192 

J^ainter^s  and  Joiner'' s  Work J  94 

(i/azier*s  Work .  194 

Jilligalioii     .     ..   .      ; 195 

Medial , 195 

Alternate 196 

I*osition 200 

Single 200 

Double ' 20  I 

Discount • 203 

iH.(l nation  of  Payments  ...........  ^' 203 

uaaging- , 205 

Mechanical  Powers '     .      .  205 

The  Lever ^ 205 

The  AxLe 206 

The  8crenv 206 

Problems ,  206 

1*^.  To  find  the  circumference  oj  a  circle  the  diameter  being  given      .  206 

2c/.   To  find  the  area  of  a  circle  the  diameter   being  given       .      ,     .  206 

^d.      To  measure  the  solidity  of  an  irregular   body ^06 

SECTION  IV. 

MISCELLANEOUS  ^ESTIONS. 

SECTION  V. 

fOllMS    OF    NOTES    ^C. 

^''^tes    .     .      .     , i     ...      2 1  I 

Bfiuds .' 212 

Recei/its 213 

Order's 214 

Deeds     ....     - 214 

Indt^nture ....f2l5 

^y^U 2  i  6 

i-XPLANAIiON     OF    THE    CHARACTERS    MADE    USE  OF    IN   THIS 

WORK. 


._    5'    The  sis^n  of  cquaiiiy  ;  as   100  c^s.ml  i^o/.  signifies  that  100  cents 
(are  equal  to  1  tiolliir. 

\.    Saint  Gkouge's  Cross,  the  sign  of  addition;  as  2-f-4=i6,  that  is 
"T"   ^.2  addtU.40  ^i  is'ct|Ua]  io  6. 
— ^The  ^s>:^  of  subtracton  ;  as  6 — 2rr-4.  ;  that  is,  2  taken  from  6  leaves  4. 

•^      Saint  ANr>uEw's  Cross,  the  sijjii  of  multiplicaiion  ;  as  4x6:zi24  ; 
^     ^  that  is,  4  limes  6  is  tqual  to  24. 

ERSED  Parenthesis,  the  sign  of  division  ;  as  3)6(2,  that  is,  6  di- 
by  3  is  equal  to  2,  or  6-f-3:p:2,. 
The  sii;n  of  proportion  ;  iis,  2  ':  4  .  :  S  :  16,  that  is,    As  2  is  to  4 
so  is  8  to  1 0 . 


THE 

SCHOLAR'S  ARITHMETIC 


<:::0«C«DO!C:5:0»000<-s 


INTRODUCTION. 


A 


RITHMETIC  is  the  art  or  scieiice  which  treats  of  numbers. 
It  is  of  two  kinds;  theoretical  and  firactical. 

TflE  TiiEoiiY  of  Arithmetic  explains  the  nature  and  quality  of  nwmbcrsand 
demonsti-ates  the  reason  of  practical  operations.  Considered  in  this  sense 
Arithmetic  is  a  Science. 

Practical  Arithmetic  shews  the  method  of  working  by  numbers,  so  as  to 
be  most  useful  and  expeditious  for  business.  In  this  sense,  Arithmetic  is  an  Art, 

Directions  to  the  Scholar. 

Deeply  impress  your  mind  with  a  sense  of  the  importance  of  arilhmefcical 
knowledge.  The  great  conccras  of  life  can  in  no  wav  be  conducted  without  it. 
Do  not,  therefore,  think  any  pains  too  great  to  be  bestowed  for  so  noble  an  end, 
Drive  far  from  you  idleness  and  sloth  ;  they  are  great  enemies  to  improyemcnt. 
Remember  that  youth,  like  th«  morning,  will  soon  be  past,  and  that  opportu- 
nities once  ncjjlected,  can  never  be  regained.  First  of  all  things,  there  must 
be  implanted  in  y©ur  mind  a  fixed  delight  is  study  ;  make  it  your  inclination  ; 
*<  j1  cienire  accemfilished  is  stveet  to  the  soul.''*  Be  not  in  a  hurry  to  get  thro* 
your  book  too  soon.  Much  instruction  may  be  given  in  these  few  words,  ww- 
derstand  every  thing  as  you  go  along.  Each  rule  is  first  to  be  com- 
mitted to  memory;  afterwards,  the  examples  in  illustration,  and,  every  r«- 
ma^k  are  to  be  perused  with  care.  There  is  not  a  word  inserted  in  this  Trea- 
tise, but  with  a  design  that  it  should  ba  studied  by  the  Scholar.  As  mucU  a« 
is  possible,  endeavor  to  do  every  thing  of  yours«lf;  on<f  thing  foimd  out  by 
your  own  thought  and  reflection,  will  be  of  more  real  use  to  you,  than  trvrnti/ 
things  told  you  by  an  Instructor.  lie  not  overcome  by  little  seeming  difncul- 
licfj,  but  rather  strive  to  overcome  such  by  patience  and  application  ;  so  shall 
your  i^rogresB  be  easy  and  the  object  of  your  endeavors  sure. 

On  entering:  upon  this  most  Useful  study,  the  first  thing  which  the  SchoUr 
has  to  rejjard  is 


.iDotation. 


Notation  is  the  art  of  expressing  numbers  hy  certain  characters  or  fijf- 
%1'es  :  ofwWich  there  arc  two  nirthods.  1.  The  Rvtnan  mcthody  by  Letters,  2. 
the  ArAbii  mct^rvd^  by  Fiywraa.     The  latter  is  lhaL«f  gcncrMi  use. 

B 


10  INTRODUCTION". 

In  the  Arabit  rncthod  all  numbers  are  expvessed  by  these  ten  charactersj- 
or  figures. 

12  34567890 

Unit, or  ;  two  ;  three  ;  four  ;    five  ;     six  ;  seven  ;  eight ;  nine  ;  cypher,  or 
one  nothing. 

The  nine  first  are  called  significa7-it figures ^  or  cf/g-zV.?,  each  of  which  standing 
by  itself  or  alone  invariably  expresses  a  particular  and'  certain  number  ;  thus, 
1  sig'iifies  one^  2  signifies  /wo,  3  signifies  three,  and  so  of  the  rest  until  you- 
come  to  nhie^  but  for  atiy  number  more  than  nhie^ix.  will  always  require  two  or 
more  of  those  fissures  set  together  in  order  to  express  that  number. 

This  will  be  more  particuhaiy  taught  by 


.IgumeiMtion. 


Numeration  teaches  ho\y  to  read  ov  write  1^x\Y  sum  or  number  by  figures*, 

In  f/ctiingdown  numbers  for  arithmetical  operations  especially  with  begin- 
ners, it  is  usual  to  begin  at  the  rii^ht  hand  and  proceed  towards  the  left. 

Example.  If  you  wish  to  write  the  sum  or  number  537,  begin  by  setting 
down  the  seven,  or  right  hand  figure,  thus,  7,  next  set  down  the  three,  at  the 
left  hand  of  the  seven,  thus  37,  and  lastly  the^^x'^',  at  the  left  hand  of  the  three, 
thus  537,  which  is  liie- number  proposed  to  be  written. 

In  this  sum  thus  written  you  are  next  to  observe  that  there  are  three  Jilacesy 
meaning  the  situations  of  the  three  different  figures,  and  that  each  of  these 
places  has  an  appropriated  !)ame.  The  ^first  filaee,  or  that  of  the  right  hand 
figure,  or  the  place  of  tlie  7  is  called  Unii^s place  ;  the  second  fdace  or  that  of 
the  figure  standing  next  to  the  right  hand  figure,  in  this  case  the  place  of  the 
3,  is  called  ten''s  fUuce  ;  th^  third  /liace,  or  next  towards  tlie  left  hand,  or  place 
of  the  5  is  called  Hundred' .h  fdace  ;  the  next  ovjourthpdace,  for  we  may  sup- 
pose more  figures  to  be  connected,  is  thousandths  place  ;.the  next  to  this  ten/t 
of  thousand's  place,  and  so  on  to  what  length  we  please,  there  being  particu- 
lar names  for  each  place.  Now  every  figure  signifies  differently,  accordingly 
as  it  may  happen  to  occupy  one  or  the  other  of  these  places. 

The  value  of  the  first  or  right  hand  figure,  or  of  the  figure  standing  in  the 
place  oi  units,  in  any  sum  or  number,  is  just  what  the  figure  expresses  stand- 
ing alone  or  by  itself  ;  but  every  other  figure  in  the  sum  or  number,  or  those 
to  the  left  hand  of  the  first  figure,  have  a  different  signification  from  their 
true  or  natural  meaning  ,  for  the  next  figure  from  the  right  hand  towards  the 
left,  or  that  figure  in  the  place  of  tens  expresses  so  many  times  te!i,  as  the 
same  figure  signifies  imits  or  ones  when  standing  alone,  that  is,  it  is  ten  timea 
its  simple,  primitive  value  ;  and  so  on,  erery  removal  from  the  right  hand 
figure  making  the  figure  thus  removed  teti  times  the  value  of  the  same  figure 
"When  standing  in  the  place  imm«diutely  preceding;  it. 

»~  .*j  J>^ 

^    fN   b 

Example.  Take  the  sum  3  3  3^  made  by  the  same  figurfe  three  time?^ 
repeated.'  The  first  or  right  hand  figure,  or  the  figure  in  the  place  o^  units. 
has  its  natural  meaning  or  the  same  meaning  as  if  standing  alone,  and  signi- 
fies three  units  or  ones  ;  but  the  same  figure  again  towards  the  left  hand  in 
the  second  place,  or  place  of  teiis,  signifies  not  three  units,  but  three  ?<??«.'?,  that 
ii^, thirty,  its  value  being  ir.cvtixscdh^cx  (rnjold proportion. ;  proceeding  on  still 
further  towards  the  left  hand,  the  next  figure  or  that  in  the  third  place,  or 
place  oi hundred's,  signifies  neither  three  nor  thirty,  but  three  hundred,  which 
i%  ten  times  the  valuer  of  ti;iit  figure,  in  the'place  immediately  preceeding  it, 
or  that  in  the  place  o^  teJis.  So  you  might  proceed  and  add  the  figure,  3,  fifty 
or  an  hundr</.d  times,  ard  everv  tiine  the  figure  wjvs  added,  it  would  sir^uifjE.' 
fen  times  more  than  it  did  tiie  last  time  before. 


INTRODUCTION.  11 

A  CYPHER  standinf^  alone  is  of  no  sii^nificalion,yet  placed  at  the  rif^ht  hand 
fdf  ungihcr  li;?ure  it  incrcaiies  the  value  ot  that  fij;urc  in  the  same  ten  fold  pro- 
.poriion,  as  if  it  had  been  preceedcd  by  any  other  figure.  Thus,  3  standing 
alone  signifies  three  ;  place  a  cypher  before  itvC^O)  and  it  no  longer  signifies 
three  but  thirty  ;  and  another  cypher  (300,)  and  it  siv^nifics  three  hundred. 

The  value  af  figures  in  conjunction  and  how  to  read  any  sum  or  number 
agreeably  to  the  foregqing  observations,  may  be  fully  understood  by  the  fol- 
lowing 

TABLE. 

.^  g                    .  The  words  at  the  head  of  the  Tabic  shew 

^^  2    •          *^  'the  sii^^nification  of  the  figures  against  whicK 

5  ■§  J  §           §  50*  they  stand  ;  and  the  figures  shtw  how  many 

«a-s^^  ig    ^       5  "«  of  that  signification  arc  meant.  Thus,  t/ri//5ia 

§  o,-  '^  I§  ^      f5  2  the  first  place  signifies  ories^  and  6  standing 

^  o  ^o^"*^':!      V  §    .  against  it  shew  that  six  onesy  or  individuals 

S-§*§      ^S"§°'  ^^'^    '^^^'®   meant  ;  re;?*  in    the  second  place 

*=  "^S  a  ?  S  ~  ?  S  §  t;  shew  that  every  figure  in  this  place  means 

,1  "^'  2  a  "§  J  J  "2  S  S  "g  «'*  t      so  niany /ews,  and  3  standing  against  it  shews 

-^  J  S  ;^  kS  S  ^  ,S  S  ^  ^  ^  jS      that  three  tens  are  here  meant,  equal  to  thirty^ 

^  ^  ^  ^  2  ^  ''^  "1  2    1   8   s'e     ^^'^^^^  ^^^^  ^-^^^'^  ^^^^'^^'  signifies.  Hundreds  in 

'TAoiAfi'^to     the  third  place  shew  the  meaning  of  figures 

'^  t  ^0  2^  0  3  7  6  A   5     ^""^'^  P^^'^^'  ^"^  be  J^undreds,   and  3   shews 

^139821064    ■^^^*  ^^^^'"''''^ ''^*'"'^^''^^*  ^^'^  "1^^"^-     I«  the  same 

2  7  o  a   1   3  6  7   5     manner  the  value  of  eacli  of  the  remaining 

4  6  5  2  7  8  9    1     ^^"''^^  ^^  the  Table  is  known.  Having  ppo- 

I   o  ^  4,  6  3  2     c^^^^^<^^  thro' in  this  way,  the  sum  of  the  first 

2  3  4.  <>  6  r     ^^^^^  °f  figures  or  those  immediately  against 

tlo9  8     '^^^^ords,  will  be  found  to  be,    Two  Bi//iom, 

T  fi    K   A     one  hundred  sixty  seven  t/tousandsy  tivo  hiai- 

o     ^^<^(^   «^<^'  thirty  Jive  Alillionfi  ;/o2ir   hundred 

twenty  one  thousands  ;  eight  hundred  and  thir.^ 

ty  six.    In  like  manner  may  be  read  all  the 

remaining  numbers  in  the  Table. 

Those  words  at  the  head  of  the  Table  are  applicable  to  any  sum  or  nun^- 
ber,  and  must  l»e  committed  perfectly  to  memory  so  as  to  be  readily  applie#< 
on  any  occasio.n.  ^ 

For  the  greater  ease  in  reckoning  it  is  convement  and  often  practised  in 
public  oflices  and  by  men  of  busines,  to  divide  any  number  into   periods  an^f 

half  periods,  as  in  the  following  manner. 


5.3   7   9,6   3  4.5    2    1,7   6   8.5    3  2, 4  6  7 

ODOeoDCo«o  looicoaseo  ojajtoorosso 

ooooo  ooooo  5:KR'»Jp^^ 

^5  §  "§  §  '^    r§  i§  §  S  ^§     §  ^  §  -^  ^  jS 

>^"^l|  >'^1"^s  -SSg^ 


12  INTRODUCTION. 

The  first  six  figures  from  the  right  hand  are  called  the  unit  fieriod^  the 
next  six,  the  TwzY^/o/i /ifi-zof/,  after  which  the  trillion^  quadrillion^  quintilliony 
periods,  Sec.  follow  in  their  order. 

Thus,  by  the  use  of  ten  fig^ure  may  be  reckoned  every  thing  which  can  be 
numbered;  things,  the  multitude  of  which  farexceeeds  the  comprehension 
ofmnn. 

"  It  may  nat  be  amiss  to  illustrate  by  a  few  examples  the  extetrt  of  num- 
"  bers,  which  are  frequently  named  without  being  attended  to.  If  a  person 
"  employed  in  telling  money  r«ckon  an  hundred  pieces  in  a  minute,  and  con- 
"  tinue  at  work  ten  hours  each  day,  he  will  take  seventeen  days  to  reckon  a 
"  million  ;  a  thousand  men  would  take  45  years  to  reckon  a  billion.  If  we 
*'  suppose  the  whole  earth  to  be  as  well  peopled  as  Britain,  and  to  have  been 
«  so  from  the  creation,  and  that  the  whole  race  of  mankind  had  constantly 
"  spent  their  time  in  telling  from  a  heap  consisting  of  a  quadiillion  of  pieces, 
<^  they  would  hai'dly  have  yet  reckoned  a  thousandth  part  of  that  quantity." 

After  having  been  able  to  read  correctly  to  his  instructor  all  the  laurabers 
in  the  foregoing  Tablt^  the  Learner  may  proceed  to  write  the  following  num- 
bers out  in  words. 


9  8 

4  3  7 

6  0  12 

7  2  8  4  5 

14  9  7  0^ 

9  7  8  3  0  16 

5   3  7  2   16  8© 


SECTION!, 


FUNDAMENTAL  RULES  OF  ARJTHMETie. 

X  HESE  are  four,  Addition,  Subtraction,  Multipli- 
cation, and  Division  ;  they  may  be  cither  simple  ox  conipoimd ; 
simple,   when  the  numbers  are  all  of  one  sort  or  denomination  ; 
"ttompound,  when  the  numbers  are  of  different  denominations. 

They  are  called,  Principal  or  Fundamental  Rules,  because  that 
all  other  rules  and  operations  in  arithmetic  are  nothing  more  than 
various  uses  and  re]>etitions  of  these  four  rules. 

The  object  of  every  arithmetical  operation,  is,  by  certain  given 
quantities  which  are  known,  to  find  out  others  which  are  unknown. 
This  cannot  be  done  but  by  changes  effected  on  the  given  num- 
bers ;  and  as  the  only  way  in  which  numbers  can  be  changed  is  ei- 
ther by  increasing  or  by  diminishing  their  quantities,  and  as  there 
•an  be  no  increase  or  diminution  of  numbers  but  by  one  or  the 
other  of  tlie  above  operations,  it  consequently  follows,  that  these 
four  rules  embrace  the  whole  art  of  Arithmetic. 

—  <^  -;.:-  -::?•  •?*>  ^;«-  ^  •»:;-  '.u-  -.u-  45*  %>  — 

§  u  .Simple  "Siftitiition.  :/ 

Simple  Addition  is  the  putting  together  of  two  or  m»re  numbers,  o(thc 
same  (lenomination,  so  as  to  make  them  one  whole  or  total  number  ;  as  3  dol« 
lars,  6  dollars,  and  8  dollars  added  or  put  together,  make  17  dollars. 

RULE. 

"Write  the  numbers  to  be  added  one  under  another,  with  units  under 
^*  units,  tens  under  tens,  and  soon.  Draw  n  line  under  the  lower  number,  then 
"add  the  right  hwid  ciilumn  ;  and  if  the  sum  be  under  tniy  write  ii  at  the 
"  foot  of  tlie  column  ;  but  if  it  be  tetiy  or  an  exact  uiimbn-  of  tens,  write  a  cy- 
^^  pher  ;  and  if  it  be  not  an  exact  number  of  tens,  write  the  excc%9  above  tent 
^'at  the  foot  of  the  column  ;  and  for  rvery  ten  the  sum  contiuni,  carry  one 
"to.the  next  column,  and  add  it  in  tlje  same  manner  as  the  farmer.  Pro- 
♦*  cecxlin  like  manner  to  add  the  other  columns  carrying  for  the  tens  of  cacK 
♦*  lo  thp  next,  aud  mark  down  the  full  sum  of  the  Icfl  hand  col\imn." 


14  SIMPLE  ADDITION,  Sec.  I.  1 


PROOF. 


l^ECKON  the  figures  from  the  top  downwards,  and  if  the  work  be  rit^ht,  th 
amount  will   be  eqiml  to  the  first  ; — or,  what  is  often   practised,  "  cut  oif  th 
"upper  line  of  figures  and  find  the  amount  of  the  rest  ;  then  if  the  amount 
<"and  upper  line  when  added  be  ecjual  to  the  sum  tQtal,  the  work  is  supposed 
*'io  be  right. 

EXAMPLES. 

i  "^  -•  d  §  "^  =.'  5'  '^*  ^  i 

t*     •-     j;   -^k  »»     "^     ?;   '^4  s     S:   ■*>* 

f!^  k5  ►<>->  I.S  t^t  >s  ^5j  ^s:  s   i,   c 

s> «!:  f^  b  ^  ti:  L-s,  b  t-^;  CN  b 

1.  What  will  be  tl>e  amount  of  S  6   1   2  dollars;  8  0  4  3  dollars  ;  6  5   1 


4 

he     ^1 


b 
dollars,  and  of  3  dollars,  when  added  together  ? 

Here  are  four  sums  given  for  addidon  ;  two  of  them  contains  wwV^,  tens^ 
Jiundreds,  i/wusands  ;  another  of  them  contains  units  tens,  hundreds  ;  and  a 
fourth  contains  unitx  only.  The  first  step  to  prepare  these  sums  for  the  op- 
eration of  addition,  is  to  write  them  down,  un^ts  under  units,  tens  under  tens, 
(^id  60  on,  as  in  the  following  manner. 

"s>ilg.5 

{      3   6   12  dollars^ 

-■^J^'-  f"".-  given  sums  for  acI-^<       g  0  4  7 doHar>. 
ditioii  plapcd  ..s  tli=  r«k  directs.-^  ^  ^    ,  ^^^^^^_ 

[_  3  doUars. 

Answer,  or  amount,    12  3  0  9  dollars^ 
Amount  of  the  thi'ee  lower  lines, 

Proof,    12   3  0  9 

To  find  the  answer  or  amount  of  the  sums  given  to  be  added,  begin  with 
du  right  hand  column,  and  say  3  to  1  is  4,  and  3  is  7,  and  2  is  9  ;  which  sum 
i(9)  being  less  than  ten,  set  down  directly  under  the  column  you  added-  Then 
■$)roceeding  to  the  next  column,  say  again  ;  5  to  4  is  9,  and  1  is  10,  being  even 
ten,  set  down  0,  and  carry  1  to  the  next  cohimn,  saying  1,  which  I  carry  to  6 
as  7,  and  0  is  nothing,  but  6  is  13  ;  which  sum  (IS)  is  an  excess  of  3  over  even 
tens;  therefore,  set  down  3  and  carry  1  for  the  10  to  8  in  the  next  column, say- 
ing 1  to  8  is  9,  and  .1  is  12  ;  this  being  the  last  column,  set  down  the  whole 
number,  (12)  placing  the  2,  or  unit  figure  directly  under  the  column,  and 
carrying  tlie  other  figure,  or  the  1,  forward  to  the  next^iiaec  on  the  left  hand, 
or  to  tliat  of  Tens  of  J'/iousands^  and  the  work  js  done.> 

It  may  now  be  required  t©  know  if  the  whole  be  right.  To  exhibit  the  meth- 
od of  prooflet  the  upper  line  of  figures  be  cut  off  as  seen  in  the  example.  Then 
adding  the  three  lawer lines  which  remain,  place  tke  amount  (8657)  under  the 
amount  first  obtained  by  the  addition  otall  the  sums,  observing  carefully  that 
each  figure  fails  directly  under  the  coluain  which  produced  it;  ^hen  m](\  this  last 
amount  to  the  upper  liiie  which  you  cut  off;  thus,  7  lo  2  is  9  ;  9  to  1  is  10  ;  care 
ry  1  to.t3  ib  7  and^G  is  iJ  ;  1  which  I  cany  again  to  G  is  9  ani  3  iu  12,  all  whick 


8 

0 

4 

3   i 

6 

5 

1   I 
3  i 

1    2 

3 

0 

9 

8 

6 

9 

7 

Sect.  I.  I  SIMPLE  ADDITION^  15 

being  set  down  in  their  proper  places,  and  as  seen  in  the  example,  compare 
the  amount  (12309)  last  obtained,  with  the  first  amount  (12309)  and  if  they 
agree,  as  h  is  seen  in  this  case  they  do,  then  the  work  is  judged  to  be  right. 

.Ve7'£.  The  reason  oi  carrying  for  ten  in  all  simple  numbers  is  evident  from 
what  has  been  taught  in  Notation.  It  is  bccaust;  10  is  an  inferior  column  is 
just  equal  in  Value  to  I  rii  a  superior  column.  As  if  a  man  should  be  holding 
in  his  right  h<wd  hdM  pisliireens,  and  in  his  left  hand,  dollars.  If  you  should 
take  10  half  pistareens  from  his  rigiit  Ikand,  and  put  one  dollar  into  Ids  left 
hand,  you  would  not  rob  the  man  of  any  of  his  money,  because  1  of  those 
pieces  in  his  Itft  hand  is  just  equal  ra  Yalue  to  \0  of  those  in  iiis  rigivt  hand. 

f 3  7  6  5  2  guineas"] 

9    A.I.W.O..*!    J ^  ^  *  ^  '^  i?"i"«as  i  so  as  to  find  the  whole  number  of 

2.  Add  togctlier<|  90153  guineas  V     . 

I  -.,       ^  «   ,        .  t^umeas 

I  2   5   3  2    1  guiHcas     ^ 


17  4  4  4  0  whole  number  of  guineas. 

13   6   7    8    8 

1    7  4  4  4  0  Proof. 

TiTE  Scholar  Who  has  given  proper  attention  to  his  rule,  and  ^he  foregoing- 
examples,  willof  liimself  be  able  to  work  the  following  ; — always  remember-- 
ing  to  carry  one  for  every  10,  and  at  the  last  column  to  set  down  the  whole 
number 


3. 

4. 

5. 

1 

5 
2 

6-  7     B 
4     0     3 
6     3     7 

2  ■ 
8 

1 

6     0     3     7 
2     4     8     0 
2     6     5      1 

3  4     7     1     2     G 
5     7     0     3     2     8 

4  2     16     8     3 

6 

7 

3 

5 

7     9 

1 

4 

2     6 

1 

7 

9 

8     7 

0 

6 

8     5 

3 

1 

2     S 

5 

8 

2 

0     4 

r> 

4 

2     0 

9 

6 

4 

5 

3 

7     6 

5 

8 

6     7 

3 

7 

3 

4 

3     2 

1 

7     S     I 


16  SIMPLE  ADDITION.  Sect.  I.  1 


10  1 


7352  S     6     S     2  26871 

3098  3  571  65734 

567  5'  942  6  83475 

1298  3678  32786 


12  13 


6 

3 

9 

8 

7 

5 

3 

4 

5 

6 

7 

8 

^ 

4 

6 

8 

2 

3 

7 

9 

8 

6 

4 

3 

5 

1 

2 

8 

7 

5 

4 

1 

7 

3 

8 

7 

9 

5 

2 

6 

7 

8 

5 

4 

0 

9 

6 

7 

4 

9 

8 

0 

I4t  15 

3  5  7  0       8  5  0  0  0  0  0 


"S^,  2  6  8  6  7  2 


1  3 


4  5  3  7  8  6  7 

9  8  9  8 

3  6  7  7 

6  0' 


Sect.  L' 1       SUPPLEMENT  to  ADDITION.  17 

Supplement  to  ^fjtlition* 


The  attentive  Scholar  who  has  understood,  and  still  carries  in  his  mind, 
what  has  already  been  taught  him  of  Ad^tion,  will  be  able  to  answer  his  In- 
structor to  the  following. 

QUESTIONS.  H 

1 .  TVha  r  is  Simfile  jiddition  > 

2.  Hoxv  do  you  f dace  number b  to  be  added  ? 
S.    Where  do  you  begin  the  jlddition  ? 

4.  How  is  the  sum  or  amount  of  each  column  to  be  set  doivn  ? 

5.  What  do  you  observe  in  regard  to  setting  down  the  sum  of  the  last  column  ? 

6.  Wiir  do  you  carry  for  10  rather  than  any  other  number  ? 

7.  How  is  Addition  firoved  ? 

8.  Of  what  use  is  Addition  ? 

JVoTE  I.  Should  the  Learner  find  any  dlffioulty  in  giving  an  answer  to 
the  above  questions,  he  is  sulvised  to  turn  back  and  consult  his  Rule,  with  its 
Illusti'utions. 

JVofE.  2,  In  treating  of  the  Rules  of  Arithnraetic  the  Scholar,  in  all  in- 
stances, i<*  not  particularly  instructed  in  the  use  and  application  of  them  to 
the  put  poses^of  life.  This  is  a  point,  however,  to  which  his  thoughts  should 
be  called  ;  tlTerC'fore  it  is  mad^a  question  here.  A  consideration  of  the  Rule 
and  of  the  quest-ions,  which  it  involves,  naturally  suggest  an  answer.  To 
cousideraiion,  therefore,  let  the  Scholar  apply  himself.  The  mind  acquires 
strength  by  exercise  ;  instruction  ought  ever  to  be  plain,  but  never  so  full 
as  to  preclude  a  necessity,  that  the  Scholar  should  in  some  degree  exercise 
his  own  thoughts  ;  it  shQuld  be  given  in  such  a  manner  as  io  force  him  into 
isome  reflections  of  his  own. 

EXEKCISES. 

1.  What  is  the  amount  of  2801  2.  Suppose  you   lend  a  neighbor 

dollars;  765  dollars;  and   of  397  /"210   at  oi>e   fime,  ^76  at  anoihev, 

dollars,  when  added  together.  ^'17    at  another,  and  £9  at  another, 

^na.  ^9Qo  dQllars.  what  is  the  sum  lejJt  ?     Ans.^212, 


A^jfE.  The  Scliolar  who  looks  at  greatness  in  his  class  wiJl  not  be  discotrr- 
aged  by  a  little  diniculty  which  may  at  first  occur  in  stating  his  (jucslion,  but 
will  apply  irnnself  the  more  closely  to  his  Rule  and  to  tijinking,  that  if  possi- 
ble he  may  be  able  ot  himsull  to  answer,  what  anotiier  muy  l)c  obliged  to  U^c 
taught  him  hy  his  Instructor. 


18 


SUPPLEMENT  to  ADDITION.     Sect.  I.  L 


3.  Wasiiingto?!  was  born 
in  the  year  of  our  Lord  1732  : 
he  was  67  years  old  when  he 
died  :  in  what  year  of  our  Lord 
did  he  die  ? 


4.  There  are  two  numbers  :  the 
less  number  is  8761,  the  difterence 
between  the  numbers  is  597  :  what 
is  the  greatest  number  ? 


5.  From  the  creation  to  the  depar- 
ture of  the  Israelites  from  Egypt  was 
2513  ye^rs  ;  to  the  siege  of  Troy,  307 
years  more  ;  to  the  building  of  Sol- 
omon's Temple,  180  years  ;  to  the 
building  of  Rome,  251  years  ;  to  the 
expulsion  of  the  kings  from  Rorne, 
244  years  ;  to  the  destruction  of  Car- 
thage, 363  years  ;  to  the  death  of 
Julius  Cajsar,  102  years  ;  to  the 
(jHiristian  sera,  44  years  ;  required 
the  time  frora  the  Creation  to  the 
Chistian  xra  ? 

ji?is.  4004  yean. 


6.  At  the  late  census,  taken 
A.  D.  1800,  the  number  of  Inhabit- 
ants in  the  JVenv- England  States  was 
as  follows,  viz.  JVew- Hampshire 
183858  ;  Massachusetts  A22%'^5  -, 
Maine^  151719  ;  Jihode  -  Island, 
69122  ;  Connecticut,  251002  ;  Ver- 
mont, 154461;  what  was  the  num- 
ber of  Inhabitants  at  that  lime  in 
JVeiu -England  ? 

Jus.  1233011  Inhabitants, 


Sect.  I.  2.         SIMPLE  SUBTRACTION.  19 

§  2.  pimple  Subtraction* 


Simple  Subtraction  is  the  taking  of  a  less  number  from  a  greater  of 
the  same  denominaiion,  so  as  to  shew  the  difference  or  remainder  ;  as  5  take;n 
from  8,  there  remains  3. 

The  greater  number  (8)  is  called  the  Mlnuendfihe  less  number  (5)  the  Sud- 
trahendj-dnd  the  difference  (3)  or  what  is  left  after  Subtraction,  the  Remai7ider, 

♦  RULE. 

"Place  the  less  number  under  the  greater,  units  under  units,  tens  under 
"tens,  and  so  on.  Draw  a  line  below:  then  begin  at  the  right  hand,  and 
"  subtract  each  figure  of  the  less  number  from  the  figure  above  it,  and  place 
"the  remainder  directly  below.  When  the  figure  in  the  lower  line  exceeds 
"  the  figure  above  it,  suppose  10  to  be  added  to  the  upper  figure  ;  but  in  thi^ 
"  case  you  must  add  1  to  the  under  figu^*e  in  the  next  column  before  you 
"  subtract  it.     This  is  culled  borroiinng  ten** 

PROOf. 

Add  the  remainder  and  subtrahend  together,  and  if  the  sum  of  them  cor- 
respond with  the  minuend,  \\\£  work  is  supposed  to  be  right. 

EXAMPLES. 

Minuend         8     6     5     3         The  numbers  being  placed  with  the  larger 

uppermosr,  as  the  rule  directs,  I  begin  with  the;- 
Subtrahend     5     2     7     1     unit  or  right  hand    figure,  in  tht  subtrahend, 

and  say  I  from  3  and  there  remain  2,  which  I 

Remainder     3     3     8     2     set  down,  and  proceeding  to  tens  or  the  next 

— — — —     figure,  I  say  7  from  i  1  cannot,  1  therefore  bor- 

Proof.  8     6     5     3     row  or  suppose    10  to  be  added  to  the  upper 

figure  (5)  which  make  15,  then  I  say,  7  from  15  and  there  remain   8,  which 

1  set  down  ;  then  proceeding  to  the  next  place,  I  say,  1  which  I  borrowed  to 

2  is  3,  and  3  fi*om  6  and  there  remain  3,  this  I  set  down,  and  in  the  next  place 
I  say  5  from  8  and  ther^  remain  3,  which  I  set  down  and  the  work  is  done. 

Proof.  I  A«d  the  remainder  to  the  ijubtrahend*  o«  finding  the  sum  just 
equal  to  the  minuend,  and  suppose  tlie  work  to  be  right. 

A*©r£.  The  reason  of  Ixfrro^ini^  ten^  will  appear  if  we  consider,  that  when 
two  numbers  are  equally  increased  by  adding  the  same  to  both,  their  difler- 
cncc  will  be  equal.  Thus  the  difference  between  3  and  5  is  2  ;  add  the  num- 
ber 10  to  each  of  these  figures  (3  and  5)  they  become  13  and  15,  still  the  dif- 
ference is  2.  When  We  proceed  as  above  directed,  we  add  or  suppose  tQ  be 
added,  10  to  !he  itimiinu!^  and  we  likewise  add  1  to  the  next  higher  place  of 
the  subtrahend^  which  is  just  equal  in  vjilue  to  10  of  tiie  lower  place. 

2.  From  3     278G532146     5  the  minuend. 
Take  I     0     6     7     9     3     6      1     2     3     4     2  the  subtrahcud 


llemaiaidcv 


29 


SIMPLE  SUBTRACTION. 


Sect.  I.  2. 


3.  From  3      16  dollars, 
Take   1     0     7  dollars. 


Remainder 
Proof 


5.  From  1 
Take  r 


0     2      3      6 
8      7     9      1 


4.  From  7     0     6     3     5  gOiiieas, 
Take        2     7     8     3  giriiieas. 


Remainder 
Proof 


7     4     2      3 
2     8     4      5 


17     9      8 
0     6     7     0 


1      0 

3      2 


Remainder  l^YVf]_dl1l_  >   T} 

^  0  zTfTTTJTJJ~~ 


6.  From  3     7     5      1  dollars,  take  16     7     4  dollars. 
Write  the  less  number  under  the  greater,  with  units  under  units,  Sec.  as 
the  rule  directs. 

7.  From  2673105,  the  minuend  ; 
take       178932,  the  subtrahend. 


Thus,  3     7     5      1 
16     7     4 


■Rem  binder 

8.  From  10000000. 
Subtract     9999999 


OPERATION. 


Minuend 
Subtralicnd 


The  distance  of  time  since  any   remarkable  eveiit,  may^be  found  by  sub- 
tractini^  the  date  thereof  from  the  present  year. 


EX. 

How  long  since  the  Amer- 
ican Indepeiultlkne,  which 
%vas  declared  in  1776. 

18     0     8  present  time 
17     7     6  date  of  the  Ind. 


Ans. 


o     3  years  since. 


So,  likewise,  the  distance  of  time  from' 
the  occurrence  of  one  thing  to  that  of  an- 
other, may  be  found  by  subtracting  the 
date  of  the  thing    first  happening,  from 
that  of  the  last.  ex. 

How  long  from  the  discovery  of  A- 
nierica  by  Columbus,  1492,  to  the  com- 
mencement of  the  war,  1775,  which 
gained  our  Independence. 

17     7     5 
14     9      2 


Ans.       3     8      S   years. 


StcT.L.2.     SUPPLEMENT  TO  SUBTRACTION.    21 

Supplement  to  ^ftttaCtiom 
QUESTIONS. 

1.  WiiAf  is  Simfile  Subtraction? 

2.  How  many  numbers  must  there  be  given  to  jierfcrm  that  ofieration  ? 

3.  Honv  must  the  given  numbers  b^  placed  ? 

4.  Wha  r  are  they  called  ? 

5.  When  the  figure  in  the  lower  number  is  greater  than  that  of  the  upfier 

mimberfrom  which  it  is  to  be  taken,  what  is  to  be  done  ? 

6.  I4ojv  does  it  appear,  that  in  sub trcK ting  a  less  number  from  a  greater,  the 

occamnally  boiTowing  often,  does  not  affect  the  diff'erence  between  these 
two  numbers  ? 

7.  How  is  subtraction  proved  ? 

8".   iViiE^y  and  honu  may  Subtraction  be  of  usv  to  a  man  engaged  in  the  pursuits 
ofHJe  P 

JEXERCISES. 

1.  What  is  the  difference  be-  2.  F.rom  a  piece  of  cloth  thatmeas- 

tween  78360  and  5421  ?  ured  691  yards,  there  were  scld  278 

Ans.  72939  yards  ;  how  many  yards  should  there 

remain  ?  ^ns.  413. 


NoxE.  In  cVLi^oHorrowifig  ten,  it  is  a  matter  ofindifrerence,  as  it  respects 
tjis;  operaiiou,  whether  we  Mipposo  ten  to  be  added  to  the  upper  figure,  and 
from  the  sum  subtract  the  lower  fij^ure  cmd  set  down  the  difference  ;  or,  as 
IVIr.  Pike  directH,  fust,  subtract  the  lower  figure  from  10,  and  adding  the 
difference  to  tJie  fij:;»re  rtbovc,  set  down  the  sum  of  tliis  difference  and  the 
upper  lij;ure.  The  l;itttr  nuthoU  may,  perha4>s,  be  thought  more  easy, 
but  it  is  conceived  that  it  docs  not  lead  the  UMdcrsiuiiding  of  youth  so  direct- 
ly ii*io  the  nature  of  the  oprr^liuu  au  the  former. 


22       SUPPLEMENT  TO  SUBTRACTION.       Sect.  1.  2. 


5.  There  are  two  numbers 
whose  diflercnce  is  3  7  5,  tire  great- 
er number  is  8  6  2  ;  I  demand  the 
less  ?  ^ir.a.  487. 


4.  What  number  is  that,  which 
taken  from  1  7  5  leaves  96  ? 
Am.     79. 


^ 


5.  Th£  capture  of  Gen.  Bur- 
coYKE  and  his  army  happened  in 
the  year,  17  7  7,  that  of  Coniwal- 
lis,  in  17  8  1?  Ijow  many  years 
between  these  events  ? 

Arts.  4  years. 


6.  Suppose  you  should  lend  a 
neighbour  2  7  6  5  dollars  at  a  cer- 
tain time,  and  he  should  pay  you 
973  at  another  ;  how  much  would 
remain  due  ?  Ans.  1792  dollars. 


7.  Supposing  a  man  to  have 
been  born  in  the  year  1745,  how 
old  was  he  in  1799  ? 

Am.  54  years. 


8.  What  nnmbcr  is  that,  to 
which  if  you  add  789  it  will  become 
6S30  I  Ans,  5561. 


§.  ^Fppo8E  a  man  to  have  been 
63  year§  old  in  the  year  1801  ^  in 
wiiul  yuar  was  he  born  ? 

•     Ana. '  In  ihc  y  car  1738. 


10.  King  Charles,  the  Martjfr 
>yas  beheaded  1684  ;  how  many 
years  since  I 


Sect.  1.  3. 


SIMPLE  MULTIPLICATION. 


23 


§  3,  ..^tmple  Jl^ultiplication. 

Simple  Mulxiplication  ttaches,  having  two  numbers  given  of  the  same 
denomiuation,  to  find  a  third  which  shall  contain  ciditM-  of  the  two  givea 
numbers  as  many  times  as  the  other  contains  a  unit. — Thus,  8  multiplied  by 
5,  or  5  times  8  is  40. — The  given  numbers  (8  and  5)  spoktn  of  together  are 
called  Factors.  Spoken  of  separately,  the  fi^-st  or  largest  number  (8)  or  num- 
ber to  be  multiplied,  is  called  the  MnltifUicand ;  the  less  number,  (5)  or  num- 
ber to  multiply  by,  is  called  the  MuUiplier  ;  and  the  amount,  (40)  the  firoduci. 

This  operation  is  nothing  else  than  the  addition  of  the  same  number  sev- 
eral times  repealttd.  If  we  mark  8,  five  times  underneath  each  oth-  8 
er,  and  add  i.hem,  the  sum  is  40,  equal  to  the  pr  v-  'net  of  ^  and  8  8 
multiplied  together.  But  as  this  kind  of  addition  is  of  frequent  8 
and  extensive  use,  in  order  to  shorten  the  operation  we  mark  down  8 
the  number  only- once,  and  conceive  it  to  be  repeated  as  often  as  8 
there  are  units  in  the  Multiplier.  — 

Before  any  progress  car.  be  made  in  this  rule,  the  following       40 
Table  must  be  committed  perfectly  to  memory. 

MULTIPLICATION  TABLE. 


ll 

2| 

3| 

4|  5\ 

6 

7 

8| 

9    1 

10  1 

iH 

'"'  1 

2 

4| 

6| 

8  1  10  1 

12  1 

14  1 

16  1 

18  1 

20  1 

22  1 

24 

3| 

6| 

9| 

12  15  1 

18 

211 

24 

27  1 

30  1 

33   1 

S6 

4| 

8| 

12  1 

16  1  20  1 

24  1 

28  1 

32  1 

36  1 

40  1 

44 

48 

5| 

10  1 

15  1 

20  1  25  1 

30  1 

35 

40  1 

45  1 

50  1 

55   1 

60 

6| 

12  1 

IM 

24  1  30  1 

36   ! 

42 

48  1 

54  1 

60  1 

66  1 

72 

7! 

U\ 

21 

28  1  35 

42 
48 

49 

56  1 

63  1 

70  1 

77  1 

84 

s| 

16  1 
18  : 

24 

32  1  40 

56 

64 

72  1 

•80  1 

88  1 

96 

9  ; 
10 

27; 

36  J  45 

54 

:  6S 

72  1 

81  i 

90  ; 

99  ; 

108 

;  20 

30 

;  40  ;  50 

60 

•70 

;  «0  ; 

90  : 

100  : 

no: 

120 

11; 

22  I 

33 

;  44  I  55 

66 

i  77 

;  88  : 

99  \ 

no  i 

12L  i 
132  i 

132 
144 

12 

124 

;  36 

;  48  i  60 

:  72 

;84 

i96: 

108  : 

12Q  ; 

By  this  Table  the  product  of  any  two  figures  ntiII  be  found  in  t  IjuI  sijuarc 
which  is  on  a  line  with  the  one  and  directly  unden*  the  olhcr.  Thus,  Su  the 
product  of  7  and  8,  will  be  found  on  a  line  with  7  «nd  tinder  8  :  so  ,2  tj|i»c.s  ':■ 
is  4  ;  3  times  3  is  9,  kc,  in  this  Way  the  Table  n\ust  be  k-jrncd  aivd  remcm 

be red. 


24 


SIMPLE  MULTIPLICATION.      Sect.  L  3, 


RULE. 

1.  Place  the  numbers  as  in  Subtraction,  the  larger  number  uppermost 
with  units  under  units,  &c.  then  draw  a  line  belqvv. 

2.  M^^HEN  the  Mul'iiiicr  does  not  exceed  12 — begin  at  the  right  Hand  of  the 
multiplicand,  and  multiply  each  figure  contained  in  it  by  the  multiplier,  set- 
tiiig  down  all  over  even  terifs  and  carrying  as  in  addition. 

3.  When  the  multifilirr  exceeds  1 2 — muiti  ply  by  each  fi.^urc  separatel v,  first 
by  the  w«i7.s- of  the  multiplier,  as  diitcted  above,  then  by  the  ^e/75,  and  the 
other  figures  in  their  order,  remembering  always,  to  place  the  first  fi;-^ure  of 
each  product  directly  un.der  the  figure  ^y  whicii  you  multiply  ;  having-  gone 
through  in  this  manner  vrivh  each  figure  in  the  JVIuUpSier,  addthcir  several 
products  together,  and  the  sum  of  them  will  be  the  product  required.       » 

,  EXAMPLES. 


1.  Multiply  5  2  9   1   by  3. 

OPERATION. 

5     2     9     1  Multiplicand. 
3  Multiplier 


8     7     3  Product. 


2.  Multiply  5  6  0  2  by  12. 

OPERATION. 
3       6       0       2 

1      2 


2     2     4 


The  numbers  being  placed  as  seen 
Vnder  the  operation,  say — 3  lines 
1  is  3,  which  set  down  directly  un- 
ci'er  the  multiplier  ;  then  3  times  9 
is  27.   set  down  7  and  carrf  2  :  again 

3  times  3  ib  6,   and  2    I  carry  is    8, 
set  down  8  ;  then  lastly,  3  tim.es  5  is 

4  5,  which  set  down   and  the  work  is 
don». 

The  numbers  being  properly 
plactd,  proceed  thus — 12  times  2 
is  24,  set  down  4  and  carry  2  ; — 12 
times  0  is  nothing,  but  2  I  carried 
is  2,  which  set  dovvn';^^ — -then  12 
limes  6  is  72,  set  down  2  and  carry 
7  ;  lasdy  12  limes  3  is  36,  and  7  I 


number, 

5.  What  is  the  product  of  4175  multiplied  by  37  ^ 
"4175  Multiplicand. 
3     7  Multiplier. 


Place  the  Factors  thus, 


2     9     2     2     5  Prod«f.t  by  the  units  (7)    of  the 

multiplier. 
2     5     2     5        Product  by  the  if72s  (3) 


15     4     4     7     5  Product  or  answer. 

\^  this  example,  as  tlve  Mu 'dpUer  exceeds  1 2,  the«pfore,  you  must  multi- 
ply hv  each  figure,  separately.  -First,  by  the  units  (7)  just  in  the  manner  of 
the  olhtr  examples.  Secondly,  by  the  tens  (3)  in  the  same  way,  excepting 
c.niv,  that  the  first  figure  of  the  product  in  the  muhiplication  by  3,  must  be 
f)h.ced  under  the  3,  that  is,  under  the  figure  by  which  you  multiply.  Lastly, 
ad'J  these  two  products  together,  the  sum  of  them  is  the  answer. 


PROOF. 
51uLTirucATio»  ir.av  be  proved  by  Division,  but  a  method  more  concrs* 


y  htm 


Sect.  I.  3.      SIMPLE  MULTIPLICATION. 


25 


and  easy,  often  practiced  by  accountants,  and  which  I  shall  recommend,  is 
called 

Casting  out  the  9'^. 

Casting  out  the  ^'s  from  any  sum  or  number,  is  the  exhausting  of  that 
number  by  the  figure  9,  till  there  is  nothing  left  of  it  but  a  remainder,  or  ex- 
cess over  even  nines  which  remainder  or  excess  is  the  thing  sought. 

How  to  cast  out  the  9V. 

Whatever  method  may  be  adopted,  this  in  effect,  is  nothing  else  than  di- 
viding the  number  by  9,  The  operation,  however,  would  be  tedious  as  nat- 
urally practised  by  division  ;  besides,  as  yet,  we  do  not  suppose  the  learner 
acquainttd  with  it.     A  shorter  and  more  successful  way  is  tlie  following 

METPIOD. 

Beginning  at  the  right  hand  of  the  number,  add  (he  figures,  and  when  the 
sum  exceeds  9,  drop  the  sum  and  begin  anew  by  adding,  first,  th€  figures, 
which  would  express  it.  Pass  by  the  nines,  and  when  the  sum  conies  out 
exactly  9,  ngglect  it ;  what  remains  after  the  last  addition  will  be  the  remain- 
der sought,  t 

EXAMPLES. 

If  it  be  required  to  cast  the  9's  out  of  576394,  proceed  thus  ; — 5  to  7  is 
12,  which  sum  (t%t}elve)  as  it  exceeds  9  you  must  drop,  and  begin  anew,  first 
add  the  figures  (12)  which  would  express  tvjelx*e^  saying  1  to  2  is  Sand 
(proceeding  with  the  other  figures,  which  remain  to  be  added)  6  is  9,  being 

•j-  This  Method  of  Casting  out  tlie  9'«  succeeds  on  a 

PRINCIPLE. 

That  every  §figure,  in  rising         ^^^^''l  """'"'^T  '^"'"  '7*^  '''•  '^r  t^""'^  '-^ 

.  Jinits  js  the    exfircssion  qj  an  individual  or 

from  the  place  of  units  to  that     ^„^^   ^^  ^j^^  ^^^^^^  ^^  ^^„^  ^j^,^  .^    -^  ^j^^  ^^_ 

of  tens,  takes  to  itself  the  addi-  pression  of  ten  individuals  or  ones  ;  there" 
lion  of  9  times  its  value.  The  fore  taking  1  (one)  its  sigvijicaucn  in  units 
same  from  tens  to  hundreds,  See.    P'^<^ce,from  10  (ten)  iis  sigwfcaiion  in  ttnt 

jilace^  leaves  9,  the  increase  0/  \y  or  9  timci 
Vsvaluc^  in  rising  from  the  place  of  units  to 
that  of  tens. 

*  A;  removed  from  unit%  filace  by  a  cufiher 
is  40,  ivhlch  divided  by  9  leaves  4  (^4  times 
9  is '^6. J 

X  6  removrd  by  a  cypher  is  60,  tvlnch  di- 
vided by  9  leaves  a  remainder  of  6  ;  cr  600 
divided  by  9  stilt  the  remainder  is  6,  the  re- 
mainder always  begins  the  same  figure  what' 
ever  may  be  the  place  of  its  removal  if  divided 
by  9. 

tt  Thus,  5683  divided  by  9,  the  remainder 
is  4  ;  let  the  figures  ivhich  express  the  num- 
ber 5G83  be  added  together — 5  to  6  is  1  \,and 
8  is  \9and  3  is  22,  ivhich  number  (2"!)  divi- 
ded by  9  leaves  a  remainder  o/"4,  tlie  same  as 
ivhen  the  number  5683  was  divided  by  9. 
Thls-e  properties  of  the  figure  9  belong  to  none  other  of  the  Digits,  cx- 
ciptingto  the  figure  3,  and  this  figure  (3)  possesses  them  inconsequence 
'•:)>-  (.'1  beinr  an  even  part  of  9. 


Consequently,  if  any  figure 
for  instance  *4  be  removed  from 
units  place  and  divided  by  9,  it 
will  leave  a  remainder  of  4  ;  the 
Mune  of  any  other  X  figure,  re- 
moved and  divided  by  9,  it  will 
leave  a  remainder  of  itself, 
aftd  that  only. 

Theuefore,  itanyttnum- 
l)..i-  be  divided  by  9  ;  or,  tlie  fig- 
ures which  express  that  number 
be  added  together  and  the  sum 
of  them  divided  by  9,  the  re- 
mainder will  be  equal. 


26  SIMPLE   MULTIPLICATION.  Sect.  I.  .3. 

eTactly  nine^  neglect  it,  and  begin  again  ;  S  to  9  twelve  ;  ngain,  drop  the  num 
(iiodve)  and  add  the  figures  (12)  which  would  express  it,  1  to  2  is  3  and  4 
is  7,  which  sum  (7)  is  the  remainder  after  the  last  addition,  or  the  thinp- 
soLiijht,  and  is  the  remainder  that  would  be  left  after  dividing  ihe  sum  57  6394 
by  b. 

To  Proi^e  Mulnplication. 

Cast  the  9's  out  of  the  MuUi/itkand  by  the  foregoing  method,  and  mark 
down  the  remainder  ;  cast  the  9's  out  of  the  Mii/d/iiier,  mark  the  remainder, 
then  multiply  tht-  remainder  first  obtained  by  this  last  remainder,  and  cast 
the  9's  out  of  the //ror/MCif  ;  also,  cast  the  9^ s  out  o^  iht  answer  ov  product  of 
the  Multiplicand  and  Muliipiicr,  then  if  these  two  last  remainders  correspond, 
j^  the  work  is  Mipposed  to  be  right. 

EXAMPLES. 

Let  7  6  6  3  0  2  be  multiplied  by  6i. 

Cast  out  the  9's  from  7  6  5  5  0  2  Remainder  5?  Remainders  multiplied 
. from  6  5  Ijtemainder  23  together. 

3  8  2   6  5    10     9's  from  10  Rem,  1  f  Corresponding 
4  5    9    18    12  ^  with  each  oth- 


er. 


9*soutof4  9  7  4  4  6  3  0  Remainder  1 


There  is  nothiiK^  more  easy  than  pi'oviiig  Multiplication  by  this  method, 
so  soon  as  the  Scholar  bhall  have  given  it  sucii  attention,  as  to  make  it  a  lil- 
xW  familiar. 

Note.  Should  the  Multiplier  or  Muliiplicand,  either  or  both,  be  less  than 
9,  they  are  to  be  taken  us  the  remainders. 

The  examples  which  follow  are  to  bt  wrought  and  proved  according  to  the 
illustrations  already  given. 

4.    Multiply     6  '  2      3     7     5  proof. 

By  8     4 


5      2      3      9      5      0     0  ProducL 

5.  Mult.  3      7     8      4     6> 

Bv  2      3      5  WVct/z^c-:,  S893910. 


Sect.  I.  3.      SIMPLE  MULTIPLICATION.  27 

6.  What  is  the  product  of  14356  multiplied  by  648  ?  Jus.  9302688. 


7.  What  i«  the  product  of  .939  £6  multiplied  by  sroi  ?  Jyis.  S  1 7793024, 


Multiply  34623217'         ,     ,  ••  ,,>^„-m«,^. 
iiy  9  6  4  8  4    J  ^'''^^'Jy  SS4058579364, 


28  SIMPLE  MULTIPLICATION.       Sect.  I.  3. 

Contractions  and  Varieties  in  Multiplication, 

Any  number  which  may  be  produced  by  the  multipliGation  of  two  or  more 
numtiCrs,  is  called  a  composite  number.  Thus  15  which  arises  from  the  mul- 
tiplicf.tiun  of  5  and  3(3  times  5  is  15)  is  a  composite  number;  and  these 
luimb^rs,  5  and  3  yre  called  Comfionent  parts.     Therefore, 

1 .  If  the  Midciplier  be  a  Composite  number — Multiply  first  by  one  of  the 
component  purls,  and  that  product  by  the  other  ;  the  last  product  will  be  thi^ 


iwer  sought. 


1.  Multiply  6  7  by  15. 

OPERATION. 

6  7 


EXAMPLES, 


5  one  of  the  component  parts. 


3  3  5 

3  the  other  component  part. 

10  0  5  Product  of  6  7  multiplied  by  15, 

2.  Multiply  367  by  48  Product,  17616. 

OPERATION. 

Consider  first,  what  two  numbers  multi- 
plied together  will  produce  48  ;  that  is,  what 
ara  the  component  parts  of  48  ?  Answer  6 
and  8(6  time?  8  is  48  )  therefore,  multiply 
367  first  by  one  of  the  component  parts,  and 
the  product  thence  arising  by  the  other  ; 
the  last  product  will  be  the  answer  sought. 


S.  Mult.  583  by  56.      Prorf,  32648.  4.  Mult.  1086  by  72.      Procf.  7819 :i 

OPERATI-ON,  OPERATION. 


2,  "  IVken  there  are  cyphers  on  the  right  hand  of  either  the  Multiplicana  or 
'^  J\^Uiplier,  or  6o//^,  neglect  these  cyphers  ;  then  place  the  significant  fig,- 
*'  ures  under  one  another,  and  multiply  by  them  only  ;  add  them  together 
*'  as  before  directed,  and  place  to  the  right  hand  as  many  cyphers  as  there  aje 
*'  in  both  the  factors." 


Sect.  1.  3.         SIMPLE  MULTIPLICATION. 


29 


EXAMPLES. 


Muliiply  65430  by  9200, 

OPERATION. 

6      5      4      S      0 

5200 


Mere  in  the  multiplication  of  65430  by 
5200,  th-e  cyphers  arc  seen  neglected,  and 
regard  paid  only  to  the  significant  fig- 
ures. To  the  product  are  prefixed  3  cy» 
pliers  equal  to  the  number  of  cyphers 
ne£?lccted  in  the  factors. 


3     4     0 


3      6     0     0     0 


2.  Mult.  3 
By 


6  5 

7  3 


0      0 
0 


3.  Mult.  78000  by  600. 

Frodiict,  46800000, 


Frod,  26645000 


3.  IVhen  there  are  cyfihers  between  the  significant fgures  of  the  ATulti filter  g 
emit  the  cyphers,  and  multiply  by  the  significant  figures  only,  placing  the 
first  figure  of  each  product  directly  under  the  figure  by  whicli  you  multiply, 
and  adding  the  products  together,  the  sura  of  them  will  be  the  product  of  the 
given  numbers, 

EXAMPLES. 


1.  Mult.  154326  by  3007, 

OPERATION. 

15  4  3   2   6 
3  0  a  7 


10   8  0  2   8  2 
4  6  2  9-78 

464058283 


In  this  example  the  cyphers  in  the  multiplier 
are  neglected,  and  154326  multiplied  only  by  7 
and  by  3,  taking  care  to  place  the  figUKe  in 
euch  product  directly  under  the  figure  frQra 
which  it  was  obtained. 


10     4     4     0      I 


30  SIMPLE  MULTIPLICATION.  Sec.  I.  3J 


3. 
48976850 
4      0      0      0      3      0 


195922O93@55O0 

4.  WhrH  the  Midlifilier  is  9,  99,  er  any  number  of9's  ;  annex  as  many  cy- 
phers to  the  MullipJicand  and  from  the  number  thus  produced  subtract  the 
multiplicand,  the   reri^ainder  will  be  the  product. 


EXAMPLES. 
I.  Mult.  6547  by  999. 

OPERATION. 

6  5  4  7  0  0  0  Write  down  the  Multiplicand,  place  as  many 

6  5  4  7  cyphers  to  the  rii^ht  hand  as   there  are  9's  in  the 

multiplier  for  a  wmw^72c/,  underneath  write  agsin 

6  5  4  0  4  5  3  the  mulliplicand  for   a   subtrahend^  subtract,  and 

the  remainder  is  the  product  of  6547  multiplied 
by  999. 


2 

3. 

6473  \ 

Pr&ducl,  640827  ' 

7021 

99S 

99 

Prod.  695079 


4. 

8     4     9     7     6  7  Proc?.  5  3844S75C24 
9     9     9     9^ 


Sect,  J.  3.     SUPPLEMENT  to  MULTIPLICATION.     31 

Supplement  to  JJlMtipCattOn. 

QUESTIONS. 

1    IV/iat  is  Simple  MuUiJilicadon  ? 

2.  //ocy  many  numbers  arc  required  to  fierforin  that  oficralion  ? 

3.  CoLLKctiYELT^  OY  together^  ivhat  are  the  givai  numbers  called  ? 

4.  SEPARAfEir  vjhat  are  thcij  called? 

5.  What  is  the  result^  or  number  sought,  called  ? 

6.  In  \:hat  order  must  t/ie  given  numbers  be  placed  Jar  multi/ilicaticn? 

7.  ffo:rdo  ijou  firoceed  when  the  Multi/ilier  is  less  than  12   ? 

8.  When  the  multiplier  exceeds  12,  rohat  is  the  method  of  procedure  ? 

9.  WnAf  is  a  composite  number  ? 

10.    What  is  to  be  understood  by  the  Component  parts  of  any  number  ? 

3  I.  How  do  you  proceed  -when  the  Multiplier  is  a  Composite  Jiumbcr  ? 

12.   When  there  are  cyphers  on  the  right  hand  of  the  multiplier^  mulliplicandy 


13.  Whet^  there  are  cyphers  between  the  significant  figures  of  the  multiplier* 

how  are  they  to  be  treated?     . 

14.  When  the  multiplier  consists  oj  9*s  how  may  the  operation  be  contracted  ? 

15.  Ho IV is  multiplication  proved  ? 

15.  By  what  method  do  you  proceed  in  casting  eut  the  9*8  from  any  number  ? 
1  7.  HoJV  is  multiplication  firoved  by  casting  out  the  9'«  ? 
18.   Of%vhat  use  is  Mulli/dication  ? 

EXERCISES. 


1.  What  sum  of  money  must 
be  divided  between  27  men  so 
that  each  may  receive  115  dol- 
lars? vc/;j*.  3105. 


J^ofE.  The  Sc!ioIar*s  business, 
in  all  questions  for  Aiiihmciical 
operations,  is  "whol'y  Avith  the 
numbeis  j^iven  ;  these  arc  never 
less  than  two  ;  tlicy  may  be  mort  ; 
and  these  numbers,  in  ont  way  or 
aviilher^  are  always  to  be  made  use 
of  lo  find'<tiie,  answer.  To  these, 
therefore,  he-  must  direct  his  at- 
tention and  carflifully  consider 
vvliat  is  proposed  by  the  question, 
to  be  known. 


52     SUPPLEMENT  to  MULTIPLICATION.    Sect.  1.  3. 

2.  An  army  of  10700  men  liavint^  5.  There  were  175  men  employed 

plundered  a  city,  took  so  much  men-  lo   finish  a  piece  of  work,    for  wliich 

ey,  that  when  it   was  sh?ired  among  each  man  was  to  receive  13  dollars? 

them,  each  man  received  46  dollars  ;  what  did  tliey  all  receive  <* 

what  was  the  sum  of  money  taken  ?  Ane,  2275. 
Jns.  492200. 


4..THERF.  is  a  ccrtc'in  Town  which  5.   If  a  man   e;?.rn  2    dollars //er 

contains  145  houses,  each   lionse  two  tveek^  how   much   will  lie  earn  in  5 

families,   and  each  family    6  Inhabi-  years,  there  being    52  weeks   in  9i 

tants  ;  how  many  are  the  Inhabitants  year?             Jns.  520  dollars. 
of  that  town.             jins.  1740. 


6.  How  m'.icb  whrat  will  3^  men  7.  If  the  p\'ice  of  wlicat  be- 1  dolbr 

thrash   in  37  days,   at  5  btishc-ls  ficr         per  bunhi-U   ^k1    4  bushels  of  wheat 
dav^  each  nu^i  ?  Ans.  6GG0  bushch.  make  1  barrel  of  Hour,  wliat  will  be 

the  price  of  175  barrels  orf'fioiir  ? 
'         Ans.  7 Q<3  dollars. 


Sect.  I.  4.  SIMPLE  DIVISION.  33 

Simple  Division  teaches,  having  iwo  numbers  given  of  the  same  denom- 
ination to  find  how  many  times  one  of  the  given  numbers  contains  the  other. 
Thus  it  may  be  required  to  knt)w  how  many  times  21  contains  7  ;  the  an- 
swer is  3  times.  ■  The  larger  number;(21)  or  number  to  be  divided,  is  called 
the  Dit^idend  ;  the  less  number  (7)  or  number'to  divide  by,  is  called,  the  Di- 
'visor  ;  and  the  answer  obtained,  (3)  the  Quotient. 

After  the  operation,  should  there  be  any  thing  left  of  the  Dividend,,  it  is 
called  the  R€mainder\  This  part,  however,  is  uncertain  ;  sometimes  there 
is  no  reminder.  When  it  does  happen  it  will  always  be  less  than  the  di- 
visor, if  the  work  be  right,  and  the  same  name  with  the  dividend. 

RULE. 
1 .  "  Assume  as  many  figures  on  the  left  hand   of  the  dividend  as  contain 
"  the  divisor  once  oroftener  ;  find  how  many  times  they  contain  it,  and  pla^e 
"  the  answer  as  the  highest  figure  of  the  quotient. 

2-  "  Multiply  the  divisor  by  the  figure  you  have  found,  and  place  the 
"  product  under  that  part  of  the  dividend  from  which  it  was  obtained. 

3.  *'  Subtract  the  product  frora  live  figures  above  it. 

4.  "  Bring  down  the  next  figure  of  the  dividend  to  the  remainder,  and  di- 
*'  vide  the  number  it  makes  up  as  before." 

When  you  have  brought  down  a  figure  to  the  remainder,  if  the  number  it 
makes  up,  be  still  less  than  the  divisor,  a  cypher  must  be  placed  in  the 
quotient,  and  another  figure  brought  down. 

EXAMPLES, 

1.  Divide  127  by  5. 
Divisor.  Dividend.  Quotient.  The  paits  in  Division  arc  to  stand 

5)127(25  thus,  the  dividend  in  the  middle,  the 

1  0  divisor  on  the  left  hand,  the  quotient 

on  the  right,  with  a   half  parenthesis 

2  7  separating  thcro  from  the  dividend. 
2   5 

^  2  Remainder. 

Proceed  in  this  operation  thus, — it  bein^  evident  that  the  divisty  (5)  can- 
not be  contained  in  the  first  figure  (1)  of  the  dividend,  therefore,  assume  the 
two  first  figures  (12)  and  enquire  liow  often  5  is  contained  in  12,  finding  it  to 
be  2  times,  place  2  in  the  quotient,  and  multiply  the  divisor  by  it,  saying  2 
limes  5  is  10,  and  place  the  sum  (10)  directly  under  12  in  the  dividend.  Sub- 
tract 10  from  12,  and  to  the  remainder  (2)  bring  down  the  n^'Xt  figure  (7)  at 
the  right  hand,  making  with  the  remainder  (2)  27.  Again  enquire  how  m*^- 
iiy  times  5  in  27  ;  5  times,  place  5  in  the  quotient,  multiply  the  divisor,i^) 
by  this  last  quotient  figure  (5)  saying,  5  limes  5  is  25,  place  the  sum  (25)  un- 
der 27,  subtract  and  the  work  is  done.  Hence  it  appears  that  127  contains, 
5,  25  limes  with  a  remainderof  2,  which  was  left  after  the  la&t  subtraction. 

This  Rule,  perhaps  at  first,  will  appear  intricate  to  the  young  Student  al- 
thotigli  it  is  attended  with  no  difficulty.  His  liability  to  errors  will  chiefly 
arise  from'  the  diversity  of  proceedings.  To  assist  his  recollection,  let  him 
notice,  that  p.  Find  how  many  limes  £<c. 

J  2.  Multiply. 
The  steps  of  division  are  fmii*^  ;>.  Subtract. 

[Ji.  BrinjfV.own. 

v\ 


34  SIMPLE  DIVISION.  Sect.  I.  4. 

It  is  sometimes  practised  to  make  a  point  (  .  )  under  the  figures  in  the  Di- 
Tidend  as  they  are  brought  down,  in  order  to  preveiit  mistakes. 

When  the  divisor  is  a  large  number  it  cannot  always  certainly  be  known 
liow  many  times  it  may  be  taken  in  the  figures  which  are  assumed  on  the  left 
Jiand  of  the  dividend  till  after  the  first  steps  in  division  are  gone  over,  but  the 
learner  must  try  no  many  times  as  his  judgment  may  best  dictate,  and  after 
he  has  multiplied,  if  the  product  be  greater  than  the  number  assumed  or  that 
number  in  which  the  divisor  is  taken,  then  it  may  always  be  known  that  the 
quotient  figure  is  too  large  ;  if  after  he  has  multiplied  and  subtracted,  the  re- 
mainder be  greater  than  the  divisor,  thea  the  quotient  figure  is  not  large  e- 
nough  ;  he  must  then  suppose  a  greater  number  of  times,  and  proceed  again. 
This  at  first  may  occasion  some  perplexity,  but  the  attentive  learner  after  some 
practice  will  generally  hit  on  the  right  number. 

2.  Let  it  be  required  to  divide  7012  by  52. 

OPERATION. 

In  this  operation  it  is  left  for  tlie  Schola-r 
to  trace  the  steps  of  procedure  without  hav- 
ing tliem  particularly  pointed  out  to  him  \jy 
words. 


Ml 

visor. 
5    2 

Dlvideyid. 
)  7  0   1   2 
5    2 

1    8    1 
1    5   6 

Quotient. 
(13  4 

2   5   2 
2  0   8 

4  4 

Retjiainder. 

TROOr. 

Division  may  be  proved  by  Mul  tiplication. 

RULE. 

«  Multiply  the  Divisor  and  Quotient  topjether,  and  add  the  remainder,  if 
«« there  be  any,  to  the  product  ;  if  the  work  be  right,  the  sum  will  be  equal 
*«  to  the  Dividend." 

Take  the  last  example. 
The  Quotient  was  134^^^^^j  j^^^^^^  j^^^^ 
1  he  Divisor  52  >  ^  ^  '^ 

268 
670 

44  Remainder  added. 

7012  Equal  to  the  Dividend. 
Another  and  more  expeditious  way  of  proving  Division  it 

By  casting  out  the  9V  W 

Cast  out  the  9's  from  the  Divisor  and  the  Quotient,  multiply  the  results* 
and  to  the  product,  add  the  remainder  if  any  :\fter  division  ;  from  the  sum  of 
these  cast  out  the  9's,  also  cast  out  the  9's  from  the  Dividend,  and  if  the  two 
last  results  agree  the  work  is  right 


Sect.  I.  4- 


SIMPLE   DIVISION, 


3.  Divide   1/354  by  86. 

OPERATION.  PROOF. 

Divisor.  Dividend,  Qjwtient.  9^^  onio^  fDivls.J   86  Bern.  5  7  Multiplied  lo- 
se)!  7  3  5  4(2  0   1     (Quot.)    201  Rem.  3  5      gether. 

1    7  2   •     •  


1   5  4 
8   6 

6  8  Rem. 

A.  Divide  153598  by  29, 

OPERATION. 
2  9)   15359S  ( 


15 


Remainder  68  added. 


9<sout  of  83i?em.2 
^<s  OHt  oi(Divid.)  \7o5^Rem.2 

.Quotient^  5296,  Rem,   14 


Agreeing 
together* 


4.  Divitle  301U  by  53»     Qtio(iS0f,  475. 


56  SIMPLE  DIVISION.  Si^ct,  1.  4, 

6.  Divide  974932  by  365.     Quotient^  2671,     Eemainder  17. 


7.  Divide  3228242  dollars  equally  among  563  men  ;  how  many  dollars 
must  each  man  receive  ?  y/«*.  5734 

From  a  view  of  the  question 
it  is  evident,  tliat  the  dollars 
must  be  divided  into  as  many 
parts  as  there  are  men  to  re- 
ceive them  ;  consequently,  the 
number  of  dollars  must  be 
made  the  dividend^  and  the 
number  of  men  the  divisor  ;  the 
quotient  will  then  shew  how 
many  dollars  each  man  must 
receive. 


Sect.  1.  4.  SIMPLE  DIVISIOK  37. 

S.  UovY  TTia»iy  tinies  does  1030603615  contuln  3215  ?    -^ns.  5.2Q561  times. 


Contractions  and  Varieties  in  Di'vision. 

1.  When  the  Divisor  does  not  exceed  V2  ;  the  operation  may  be  performed 
without  setting  clown  any  hsjures  excepting  the  quotient,  by  carrying  the 
computation  in  the  mind.  The  units  which  would  remain  after  subtracting 
the  prcduct  of  the  quotient  figure  and  the  divisor  from  the  figures  assumed 
of  the  dividend,  must  be  accounted  so  many  tens,  and  be  supposed  to  stand  at 
the  left  hand  of  the  next  figure  in  the  dividend,  then  consider  again  how 
often  the  divisor  may  be  had  in  the  sum  of  them.  Proceed  in  tliis  way  till  all 
the  figures  in  the  dividend  have  been  divided.  This  is  called  short  division. 

EXAMPLES. 

1.  Divide  732  by  3 

OPEllATIOX. 

3  )7  3  2(  244  HeU'E  I"  say,  h'ow  often  3'in  7;  knowing 

it  to  be  2    times,   I  place  2  in  the  quotient, 
then  considering   that  the   quotient   figure 
(2)  and  the  divisor  (3)  multiplied  togelher 
would  be  6,  and  that  this  product  (6)  sub- 
tracted from  7,  in  the  dividend,  would  leave  1, 1  then  consider  this  remainder 
(1)  as  standing  at  the  left  hand  of  the  next  figure  (3)  of  the  dividend  which 
together  make  13.     I  now  say,  how  many  times  3  in  13 — 4  limes,  therefore, 
I  i)lace  4  in   the  q.uolient  whicli  multiplied  into  the  divisor  (3)  would  be  12, 
and  12  subtracted  from  13  would  leave  ),'  which   considered  as  standing  at 
the  left  hand  of  the  next  or  last  figure  (2)  of  the  dividend  would  uxakv:  12,  a- 
gain  how  many  times  (3)  in  12-r-4  tinges, — I  then  ]ilace  4  in   the  quotient, 
which  multiplied  into  the  divisor  (3)    is  12,  this  product  (12)   I.  consider  as 
subtracted  -from  12, 1  find  there  will  i)e  no  remainder,  and  tlie  work  is  done. 

Note.  The  quoiieiU  may  stand  »is  it  is  seen  in  the  cxumi^lQ,  or  it  may  be 
placed  under  the  dividend,  thus, 

3   ) 

2  4  4 


SIMPLE  DIVISION.  Sect.  1 


n 


2.  Divide  37426  by  7.  Here   I  say,   how  often  7  in   37  ?  5 

OPERATION-  times  and  2  remain  ;  then,  how  oftea 

7)37426  7in24?3  times,  and  3  remain  ;  how 

often  7  in  32  ?  4  times  and  4  remain, 

Quotient  5  3  4  6  Mem.  4.  lastly  how  often  7  in  46  ?  6  times  an4 

4  remain. 


3.  Divide  12363  by  5.    Quot.  2473.  Rem.  3. 


Divide  602571  by  8.     Quot.  75321  Bern.  3, 


II.  When  there  are  cyfihers  at  the  right  hand  of  the  Divisor,  cut  them  off 
also,  cut  off  an  equal  number  of  figures  from  the  right  hand  of  the  divi- 
cbnd  and  place  tliesj?  figures  at  tlie  right  hand  of  the  remainder. 

EXAMPLES. 
•  1.  Divide  6203916  by  5700. 

OPERATION. 

57  I  00)  62039  [  45  (1088  Here  are  two  cyphers  on  the  rif^ht 

57  •  •  •  hand  oi   the  divisor,  which  I  cut  off; 

also,   I  cut   off  two  figures  (46)  from 

503  the  dividend,  and  to  the  right  hand  of 

456  the   remainder  after   the  last  division 

(23)1  place  the   figures  cut  off  from 

479  the   dividend    (46)   which    make    the 

4*5  whole  remainder  2346. 


SSdG  Re:n. 


&£CT.  I.  4.  SIMPLE  DIVISION.  29 

2.  Divide  379452  by  6500.     QuoL  5S.  Rem.  2432, 


3.  Diyide  2764503721  by  83000.      Quot.  33307.  Rem,  22f21, 


III.  IV/icn  the  divisor  is  10,  100,  1000  or  1,  imth  any  number  ofcyfiliert  zri' 
nexedi  cut  off' as  many  figures  on  the  right  hand  of  the  dividend  as  there  are 
cyphers  in  the  divisor  ;  the  figures  which  remain  of  the  dividend  compose 
the  quotient ;  those  cut  off,  the  remainder. 

EXAMPLES. 

1.  Divide  1576  by  10.  Here  We  have  one  cypher  in  the  divi- 
OPKUATION.  sor  ;  therefore,  cut  off  one  figure  (6)  from 
1  I  0)  1   5  7  I  6                        the  dividend  ;  what   remains,  (157)  is  the 

quotient  and  the  figure  cut  off  (^6)  the  re- 
mainder. 

2.  Divide   3217  by   100. 

OPERATION. 


Quot.  Rem, 
I  d  0)3  3  I  I  7 


40  SUPPLEMENT  TO  DIVISION,  S£ct.   14. 

Supplement  to  ^jljj^ion^ 

QUESTIONS. 

1.  WHAfi.i  Si7n/ile  Division  ? 

2.  How  many  numbers  must  there  be  given  to  Jierform  that  operation  ? 

3.  What  are   the  given  numbers  called? 

4.  Hojr  are  they  to  stand  for  Division  ? 

5.  How  many  stefis  are  therein  Division? 

6.  What  is  the  first  ?  the  second  ?   the  third  ?  the  fourth  ? 

7 .  WnAf  is  the  result  or  answer  called  ? 

8.  Is  there  any  other,  or  uncertain  fiart  pertaining  to  Division  ?     What  is  it 

called  ? 

9.  Of  what  name  or  kind  is  the  remainder  ? 

10.  IVHAfis  short  Division? 

11.  When  there  are  cyphers  at  the  right   hand  of  the  Divisor,  ivhat  is  to  be 

done  ? 

12.  What  do   you  do  noith  fgures  cut  off  from   the  Dividend  when  there  are 

cyphers  cut  off  from  the  Divisor  ? 

13.  When  the  Divisor  is  10,  100,  or  1  with  any  number  of  cyphers  annexed^ 

how  may  the  operation  be  contracted. 

14.  How  niany  ways  may  Division  be  proved  ? 

15.  How  is  Division  proved  by  Multiplicatio^i  ? 

16.  How  may  Division  be  proved  by  casting  out  the  9'«  .? 

17.  Of  whai"  use  is  Diviso7i  ? 

EXERCISES. 

1.  Suppose  an  Estate  of  36582  dol-  2.   Ax  army   15000  raen  havin<^ 

lars  to  be  divided  among  13  sons,  how  plundered  a  city,  took  2625000  dol- 

iT2ucli-Sj»rould  each  one  receive  ?  iars  ;  what  \vas  each  man's  share  ? 

Jns.  'ISl^^'cfbliars.  Ans.  175  dollcrrs. 


Sect.  I.  4.  SUPPLEMENT  to  DIVISION.  41 

3.  A  certain  number  of  ri.en  were  4.  If  7412  eggs  be  packed  in  54 

concerned  in  the  payment  of  18950         casks,  how  many  in  a  cask  ? 
dollars,   and  each  man  paid   25  dol-  Jns,  218. 

lars,  what  was  the  number  of  men  ? 
A71S.  758. 


5.  A  farm  of  375  acres  is  let  for  6.   A  field  of  27  acres  produces 

1125  dollars,  how  much  docs  it  pay         675  bushels  of  wheat;  how  much 
per  acre  ?    Am.  3  dollars.  is  that  per  acre  ?      Ans.  25  buahela 


7.  Suppposing  a  man's  income  to  8.  What  numbier  must  I  multiply 

be  2555  dollars  a  year  :  how  much  by  13,  that  the  product  may  be  371  ? 

is   that  per   day,   thtre  being   365  Ans,  67, 
days  in  a  years  ?             Am.  7  dollars . 


F 


42  COMPOUND  ADDITION.  Sect.  1. 5, 

§  5.  Compountr  ^tJtiition. 


Compound  Addition  is  the  adding  of  numbers,  which  consist  of  articles 
of  different  value,  as  pounds,  shillings,  pence,  and  farthings,  called  (liferent 
denominations  ;  the  operations  are  to  be  regulated  by  the  value  of  the  articles 
which  must  be  learned  from  the  Tables. 

RULE    FOR    COMPOUND    ADDITION. 

1.  Place  the  numbers  so  that  those  of  the  same  denomination  may  stand 
directly  under  each  other. 

2.  Add  the  first  column  or  denomination  together,  and  carry  for  that  num- 
ber which  it  takes  for  the  same  denomination  to  make  1  of  the  next  hi""her. 
Proceed  in  this  manner  with  all  the  columns,  tili  you  come  to  the  last,  which 
must  be   added  as  in  Simple  Addition. 

1.  OF  MONEY. 

TABLE. 

4  FartTiings  <7r."J  f  Penny,  marked  d. 

12  Pence  v make  one-}  Shilling,  &. 

20  Shillings        J  (^ Pound,"  ^'. 

EXAMPLES. 

1.  What  is   the  sum  of  ^61       17s.      5d. £\Z     3«.     Sc/. and  of 

^S     166'.     \\d,     when  added  together  ? 

OPERATION. 

^.      s.      dr\ 

16       17         5 


I      Those  numbers  of  the  same  denomination 
13         3         8  fplaced  under  each  other,  as  the  rule  directs 

"J 


6 


30        "18  0 

I  begin  with  the  right  hand  cokimn  or  that  of  pence,  and  having  added  it, 
find  the  sum  of  the  numbers  therein  contained  to  be  24;  now  as  12  of  this 
denominaiion  make  one  of  the  next  higher,  or  in  other  words,  12  pence  make 
one  shilling,  therefore  in  this,  or  in  the  column  of  pence  I  must  carry  for  12  ; 
I  now  enquire  how  often  12  is  contained  in  24,  the  sum  of  the  first  column  or 
that  of  pence  ;  knowing  it  to  be  2  times  and  nothing  over,  I  set  down  0  un- 
der the  column  of  pence,  and  carry  2  to  that  of  shillings,  to  be  added  into 
the  second  column,  saying  2  I  carry  to  6  is  8  and  3  is  1 1  and  7  is  18  and 
10  lo  18  is  S8  and  10  aguin  is  38  (for  so  each  figure  in  tens  place  must  be 
reckoned,  I  in  t])at  place  being  equal  in  value  to  10  units.)  Now  as  20  shillings 
make  one  pound,  therefore,  in  the  column  of  shillings,  1  carry  for  2®  ;  I  then 
enquire,  .how  often  20  in  38  i  once,  and  18  remains,  therefore,  1  set  down  di- 
rectly under  the  column  of  shillings  .18,  what  38  contains  more  than  20,  and 
for  the  even  20  carry  1  to  pr^unds  or  the  lu:st  column,  which  is  to  be  added 
after  the  manner  of  Simple  Adclition. 

Note.  Thrt  nethod  of  proof  for  compound  addiuon  is  the  bame  as  that 
of  bimple  addition. 


Sect.  I.  5. 


COMPOUND  ADDITION. 


43 


1  8  4 

2  6      15 
8  1 


d. 

1     1 

3 

7 


qr, 

0 
3 


S3    /    10    0 


3  7   1 

5 

6   8 


1    5 

7 


d.  qr, 

6  2 

4  O 

2  1 


^^1        :^6  6  Z 


4.  Supposing  a  man  goes  a  journey,  and  on  the  first  day 

1802             May   14,  Pays  for  a  dinner £0  1  6 

ior  oats  for  his  horse  -  -  -  0  0  6 

- — for    sling  ---------o  1  2 

15     for  supper  and  lodging  -  -  0  2  0 

for  horse  keeping 0  1  10 

for  bitters 0  1  6 

for  breakfast --0  2  0 

to  the  barber  for  dessing  -  0  I  6 

---for  dinner  again  and  other 

refreshment            0  S  5 


What  were  the  gentleman's  expences  ?  0  /  D       5 

5.  Suppose  I  am  indebted  £.  «.        rf. 

To  A.  Thirty  iW9 /Kunds, fourteen  shillings  and  ten  fience.*J^  '%' 

— 1^.  Forty  one  fiounds^  six  shillingSy  and  eight  fience,  -y  /  ^ 

— C.  Seventy  Jive  fiounds,  eight  shillings  ^  /J  O 

— D.   Three fiounds^  and  nine  p,ence.  A  y 

How  mucii  is  the  debt  ? 


-^-   /S2   10    ^ 


6.  A  MAN  purchases  cattle  ;  one  yoke  of  oxen  for  C\^  116;  four  cows 
for  ^  18  19  7  ;  and  other  stock  to  the  amount  of  ;C2|  t/T  what  was  the 
araouiit  of  the  cattle  purchased  ?  Ans.  £,s\   16  1. 


U  COMPOUND  ADDITION.  Sect.  1.  5, 

OF  TROY  WEIGHT. 
By  Troy  weight  ate  weighed  gold,  silver,  jewels,  electuarios  and  liquors. 

TABLE. 


24  Grains  5^r*.  "J  f  Penny  weight  marA-tfrf  pwt. 

20  Pennyweight  >  make  one  ^  Ounce  oz. 

12  Ounces  J  /  Pound  lb. 


EXAMPLES. 

1 
lb.  oz.  pwts.  grs. 

7  0  10  13  4  Because  24  grains  make   a 

3  9  7  16         pennyweight,  you  carry  one  to 

2  8  ©  0  5         the  pennyweight  column  for  ev- 

-^  3  6  2         ery  24  in  the  sum  of  the  column 

f/\/y'^        t    t  " SC" "Pj        of  grains  :    because  20   pcr^y- 

lOy            II                 7              lj7        weights  make  1  ounce,  you  carry 
—  4> ^ for  20  in  pennyweights,  and  be- 
cause  12  ounces  make   1  pound 

■  — —        you  carry  tor    12  in  the  ounces. 

This  is  called  carrying  accord- 
ing to  the  value  of  th«  higher 
place. 


2.  3. 

lb.' 


lb. 

oz. 

pwt. 

1  6   1 

7 

1   9 

6 

5 

6 

2  8 

0 

1  4 

3 

7 

19  6 

s 

6 

3^ 

f 

7 

oz. 

7 

pwts. 
1   4 

2  S 

2 

0 

6 

1  1 

1  3 

5 

1  0 

1  2 

7 

53 

^.. 

/ 

U 

/ 

y^ 

/y/^       S      [y  35       S       J 

Note.  The  fineness  of  gold  is  tried  by  fire  and  is  reckoned  in  ccra^*,  by 
which  is  understood  the  24th  part  of  any  quantity  ;  if  it  lose  nothin«g  in  the 
trial,  it  is  said  to  be  24  carats  fine  ;  if  it  lose  2  carats,  it  is  then  22  carats  fiiie, 
which  is  the  standard  for  gold. 

Silver  which  abides  the  fire  without  loss  is  said  to  be  12  ounces  fine. 
The  standard  for  siver  coin  is  1 1  oz.  2  pwts.  of  fine  silver,  and  IS  pwts.  of 
eopper  melted  together. 


Sect.  1.  5.  COMPOUND  ADDITION.  45 

3.  OF  AVOIRDUPOIS  WEIGHT. 

By  Avoirdupois  weight  are  weighed  all  things  of  a  coarse  and  drossy  nature, 
as  tea,  sugar,  bread,  flour,  tallow,  hay,  leather,  and  all  kind  of  metals,  except 
gold  and  silver. 

TABLE. 

16  Drams  dr,  "^  f Ounce,     marked,  oz, 

16  Ounces  |  |  Pound,         —  lb. 

2g  Pounds  J>make  one<^  Quarter  of  a  hundred  weight,  yr. 

4  Quarters  |  |  100  weight,  or  1 12  pound,      civt, 

20  Hundred  weight  J  L'^'on,  —  —  T, 

EXAMPLES. 

I 
T.  civt.  qr.  lb.  tz.  dr. 

186             3              225  11  8 

417023  7  6 

9837  2  5 

2             3              1«16  5  II 

toz   /3      0     77?        Id     7^ 


2. 

T. 

cwt. 

yr. 

lb. 

ez. 

«^. 

8     0     1 

3 

2 

2      5 

1      I 

8 

7 

1     9 

3 

1     4 

5 

^ 

8     6 

2 

0 

6 

0 

1      5 

3 

7 

1 

0 

6 

i 

9  9  (? 

/t 

3 

Itf 

(1^ 

/ 

9  r 

? 

0 

10 

/i 

y 

8i  d      rL       s       /s         8       / 

Note.  "  175  Troy  Ounces  are  precisely  equal  to  192  Avoirdupois  Ounces, 
.1  to  U4  Avoirdupois.  1  lb  Troy=:5760  y;rain», 


and  175  Troy  pounds  are  equa 
Mud  I  lb  Avoirdupoisiz:7000  grains 


/' 


46  COMPOUND   ADDITION.  Sect.  1.  5 


4.  OF 

TIME; 

TABLE. 

• 

60 

Seconds  9. 

•" 

f  Minute,  marked^    m. 

60 

Minute* 

1  Hour, 

h. 

24 
7 

Hours 
Days 

i>makeone^D^Jj^^ 

d. 

IV. 

4 

Weeks 

1 

1  Month, 

mo. 

13 

Months  kf.  if 

U.J 

1^* Julian  year. 

Y. 

PVAMPLES. 

r. 

mo. 

«>. 

u 

h. 

m. 

». 

1  6 

1    0 

3 

6 

2  3 

5  7 

4  3 

2^8 

7 

2 

5 

1   6 

2  8 

3  2 

3  9 

6 

1 

3 

1  7 

3  8 

1   1 

8  7 

4 

0 

1 

1  4 

1  5 

1  7 

i/i 

3 

0 

u 

0 

/^ 

'/y^ 

ISS 

iS- 

0 

^ 

0 

2^ 

^ 

/7X     :3      ^     4^        ^      /^     ^.3 


F.  77219.  W.  </.  A.  7M.  *. 

89    11      3     6  22  45    36 

3610      2     5  6  5544 

87      2      1     0  11  2233 

36433  5  8      7 

%^0  3  3  /  ZZ  IZ.  0 
160  ^  3  /  3.3  X6  Z4/ 
X?p         3        5      /         ZL       7X         ^ 

The  number  of  days  in  each  Calendar  month  may  be  remembered  by  the 
foiiowing  verse. 

THiR'Tr  days  hath  Sgfitemb<;rj  jljiril^  June  and  Kov ember  : 
February^  tiventy-eight  alone  ;  ail  the  rest  have  thirty'One. 

"  *The  civil  Solar  year  of  365  days  being  short  ©f  the  true  by  5h.  48m. 
57s.  occasioned  the  beginning  of  the  year  to  run  forward  thro'the  season  near- 
ly one  day  in  four  years  ;  on  this  account  Julius  Caesar  ordained  that  one  day 
should  be  addpd  to  February  every  fourth  year,  by  causing  the  24th  day  to  be 
r-eckoned  twice  ;  and  because  this  24th  day  was  the  sixth  (sextiliis)  before  the 
kalends  of  March,  tliere  were,  in  this  year,  two  of  these  sextiles,  which  gave 
the  name  of  Bissextile  to  this  year,  which  being  thus  corrected,  was,  fiom 
thence,  called  the  Julian  year. 


Sect.  I.  5. 


COMPOUND  ADDITION. 


47 


60  Seconds 
60  Minutes 
30  Degrees 
12  Signs,  or  360  de- 
grees 


5.  OF  MOTION. 
TABLE. 

"^  f  Prime  minute,  marked    "  ' 

I  I  Degree 

[>make  one<J  Sign  s. 

J  J    C  The  whole  great  circle 

L  ^  of  the  Zodiac. 


EXAMPLES, 


2  5 

1  7 

6 

1  0 


17  18 

4  9  S  ^ 

^   S  2  4 

17  16 


S9    S9_SJi 

^9        ^      ^^ 


V 


8 
2  6 
i  8 

9 


S  S  4  4 

4  4  S   S 

3  6  12 

Z  ^  2  2 


11     3    SO    73 


II    U    ^^ 


JT^ 


19 
T3 


6.  OF  CLOTH  MEASURE. 


TABLE. 

2  Inches,  one  fifth  /«.                          "]  fNail,  marked 
4  Nails,  or  9  inches  Quarter  of  a  yard, 

4  Quarters  of  a  yai  \  or  36  inches  Yard, 

3  Quarters  of  a  yard,  or  27  inches  Ell-Flemish, 

5  Quarters  of  a  yard,  or  45  inches    J>make  one<J  Eil-English, 

6  Quarters  of  a  yard,  or  54  inches  Ell-«French, 

4  Quarters  1  inch  and  one  fifth, or  ..,,  ^        , 

37  inches  and  one  fifth  i.ii-acoicn, 

3  Quarters  and  two  thirds  J  [^Spanish  Var. 


na, 
qr, 
yd. 

E.  FL 
E.E, 

E.  Er. 

E,  Sc, 


EXAMPLES, 


r^s. 

1  4 

3  6 
7 
1 

1  5 


6:^r 


qr. 

3 

1 
0 
2 
3 


^  0 


3    I 


:->     o 


E.  E, 
1  9 

5  6 
7 

6  3 
1  8 


/^.f 


2. 
qr. 

3 
1 

2 
0 
2 


/{^S-       0 


\JS     0      0 


J      ^ 


o 


^v3 


COMPOUND  ADDITION. 


Sect.  1.  5. 


7.  OF  LONG   MEASURE. 


Bv  Lonj^  Treasure  are  raeasured  distances,  or  any  thing  where  length  is 
con^iderfcd  Avithout  regard  to  breadth. 

TABLE. 


3  Barley  corns 

bar.        "^ 

rinch           marked 

m. 

I'i   Inches 

Foot, 

/^ 

3   Feet 

Yard. 

VcT. 

5-^  Y^rds,  or  16.V 

feet 

Rod,  Perch,  or  Pole,/zo/. 

40  Poles 

^make  one<j 

Furlong, 

>r. 

8  Furlongs 

Mile, 
C  Degree  of  a 

Thile 

691  Statute  miles 

nearly 

I  great  Circle 
^  A  great  Circle 

deif. 

30O  Degree* 

-< 

L  ^  of  the  Earth. 

EXAMPLES 

• 

Drg:               ml. 

>r. 

1- 

ft.               in. 

bar 

16  8         5  7 

7 

2  6 

15            11 

2 

12  4          5  3 

6 

1  8 

7             6 

1 

7   9          3   6 

1 

7 

9          1  0 

0 

4              7 

3 

0 

3              2 

1 

^;/   /^  ^ 

13 

3 

6    1 

^^(^     l/i  z. 

16 

3i 

6    Z 

d77 


/6 


Z. 


/5 


/ 


2. 


Brg. 

wf. 

fur. 

fioL 

Z'^. 

in. 

1   3 

5  6 

5 

1   3 

8 

1 

4  9 

1   8 

1 

2  7 

1  6 

Q 

2  6   7 

1   2 

3 

1  6 

9 

0 

2   9 

8 

0 

5 

3 

1 

J^^ 

2St 

^ 

i^3 

3 

4^ 

■  j^r 

3S> 

r 

^ 

J/t 

.3 

J>"^      '^"rt    L    -  Z3     ~~^3       ?■ 


Sect.   I.  S.  COMPOUND  ADDITION. 

8.  OF  LAND  OR  SQUARE  MEASURE. 
By  Square  Measure  are  measured  all  things  r.lvat  have  length  and  breadth. 

TABLE. 


144  Inches 

Lmake 

! 

fSqu 

are  foot, 

9  Feet 

-Yard, 

30\  yards,  or") 
272-L  Feet         3 
40  Poles 

one<J 

-Pole, 
-Rood, 

4  Roods,or  160  Rods,  > 
or  4840  yards       3 

/^     _ 

-Acre, 

640   Acres 

J 

L — 

-Mile. 

EXAMPLES. 

Jcres.               rood. 

poL 

Jt. 

i?i. 

3   7  6              3 

3  6 

9  3 

1 

2   I 

5  6  8              1 

2  7 

5  8 

7  6 

2  4  7             2 

3  5 

6   I 

2  4 

7//J      0 

/^' 

lid 

;/ 

816       0 

ix 

in 

^'^ 

//9S  0         IS        213  // 

9.  OF  SOLID  MEASURE. 

By  Solid  Measure  are  measured  all  things  that  have  length,  breadth,  and 
thickness. 

TABLE. 

1728  inches  f  fFoot. 

27  Feet  |  |  Yard. 

40  Feet  of  round  tinxjjer,  or  >    J  make  one  J  r^ r  r.^,\ 

50  feet  of  hewn  limber  S     \  ?   ^""  °'  ^''^^' 

128  Solid  feet,  i.e.  Sin  leuirth?  \  r-     ^    e  ^xt      \ 

4  in  breadth,  &  4  in  heighi  \  J  [^^^^  ^^  ^^°°^- 

EXAMPLES, 
1.  2. 

Ton.  y?.  z«.  Con^  ft.  in. 

65  37  229    39  118  1021 

19  2  6  1207      3  SO  437 

36  17  54    18  72  65  9 

57  38  629  86  124 

W     Jd~~JW6     Tt       7/      SL^ 

W     0     lu;    SI      sb~rito 

JTl     To       VJU      Tf       TT       'TT'^ 

G 


so 


COMPOUND  ADDITION, 


Sicr.  I.  5. 


10.  OF  WINE  MEASURE 

By  Wine  measure   are  measured  Rum,  Brandy,  Perry,  Cyder,  Mead, 
Vinegar  and  Oil. 

TABLE. 


2  Pints        fits. 

r        1 

Quart, 

marked 

qts. 

4  Quarts 

Gallon, 

gal' 

10  Galk)ns 

Anchor 

of  Brandy, 

arte. 

18  Gallons 

■»- 

Runlet, 

ru?i. 

,1^  Gallons 

< 

make  one<|  Haifa  1 

tiogshead 

Ihhd. 

42  Gallons 

Tierce, 

tier. 

63  Gallons 

Hogshead, 

hhd. 

2  Hogsheads 

Pipe  or 

Butt,        P. 

otB. 

2  Pipes 

J 

Ltun, 

T, 

EXAMPLE 
1. 

Hhd. 

gal. 

y^*. 

fits. 

3  9 

5  2 

3 

I 

1   6 

2  7 

1 

0 

3   5 

1  2 

0 

1 

2  9 

5  8 

2 

0 

nil 

^30 

8t 

\U     3      1 

T.              hhd. 
^   S                2 
S  5              1 

1  7              0 

2  3              2 

^        5        0 

gal.              qts.               fit*. 

5  8              3              1 
3  Q              1              0 
2  9             2              1 
12              1              0 

//^      / 

II      0     0 

77     0 

IS     0     / 

114        I 


II 


0 


0 


N.  B.  A  PINT  ^vine  meaaui'c,  is  28J  cubic  inches. 


Sect,  1.  5. 


COMPOUND  ADDITION. 


51 


"2  Pints 
4  Quarts 

8  Gallons 
8|-  Gallons 

9  Gallons 
2  Firkins 
2  Kilderkins 

li  Barrel,  or  54  Gallons 
2  Barrels 
;3  Barrels,or  2  hogshead^ 


Jl.  OF  ALE  OR  BEER  MEASURE. 

TABLE. 

Quart,               marked  9?*- 

Gallon,  gal- 

Firkin  of  Ale  in  London,  A.  fir. 
Firkin  of  Ale  or  Beer, 
•i  makeonc<J  Firkin  of  Beer  in  London,  B/r. 

j  Kilderkin,  Kill. 

Barrel,  ^ar. 

I  Hogshead  of  Beer,  hhd. 

I  Puncheon,  Pun, 

LButt,  -Bw«- 


Hhd. 

2  8 

17  3 

2  7 


gal. 

4  8 

5  0 
2  4 
1   6 


EXAMPLES. 


9. 


B.fr. 

2  3 
4  5 
9  8 

3  6 


^2^ 

7T7" 


:^/  0 


)/    z. 


K.  B.  A  PINT,  Beer  measure,  is  35{-  cubic  inches. 


.2. 

g-fl/. 
6 
2 
7 
8 


/7^         ^ 


3 
1 

0 


SST^  3/  Z       Zor  6  ^ 


/JV     0    0 


:9 


6.  OF  DRY  MEASURE. 

By  Dry  Measure  are  measured  all  dry  goods,  such  ^s  Corn,  Wheat,  Seed 
Fruit,  Roots,  Salts  Coal,  8cc. 


2  Pints 
2  Quarts 
2  Pottles 
2  Gallons 
4.  Pecks 

2  Bushels 

3  Strikes 
2  Coonis 

4  Quarters 
4^  Quarters 

5  Quarters 
•2  Weys 


TABLE. 


'Qr.art,             marked 

7f*. 

Pottle, 

pot. 

Ciallon, 

gaU 

Peck, 

pk. 

Bushel, 

hM, 

^makc  one<;  Strike, 

9tr. 

Goora, 

CO. 

Quarter, 

yr. 

. 

(Jhaldion, 

ch. 

Clialdron  in  London, 

Wey, 

wy. 

\ 

LLast, 

M*(. 

53 

COM 

1. 

POU> 

.^D   ADDITION. 

EXAMPLES. 

2. 

Sj 

ECT. 

1.  5. 

Bufi, 

ilk. 

qta. 

ptB. 

Ch. 

tus: 

/2^. 

9?*. 

2  7 

2 

6 

1 

3  7 

1   6 

2 

5 

1   8 

3 

7 

0 

2  6 

2  8 

3 

7 

2  0 

0 

1 

1 

1  8 

1   5 

1 

0 

19 

1          3          0 

t\> 

^7    ^  ^ 

/i? 

/    3.   1 

17         2  5 


irr     i^    '} 


i 


N.  B.  A  GALLON,  Dry  Measure,  contains  268  ^  cubic  inches. 
The  folloxmn^  are  denominations  of  things  counted  by  //^  Table. 

*  12  Particular  things  raake       1  Dozer, 

12  Dozen  1   Gross, 

12  Gross  or  144  dozen  great  Gross. 

ALSO 

20  Particular  things  make  one  Score. 

Denominations  of  measure  not  included  in  the  Tables. 

6  Points  make     I    Line, 

12  Lines       —  Inch, 

4  Inches         —        Hand, 

3  Hands       —  Foot, 

66  I'\;et,  or  4  Pole^a  Gunter's    Chain, 

3  Miles  League. 

A  Hand  is  used  tp  measure  Horses A  Fathom,  to  measure  depth*.— 

A  League,  in  reckoning  distances  at  Sea. 

N.  B.  A  Quintal  of  Fish  weighs  I  Cwt.  Avoirdupois. 


Sect.  I.  6. 


COMPOUND  SUBTRACTION. 


^ 


§  4.  Compound  ^Subtraction, 

Compound  Subtraction  teaches  to  fuid  the  difference  between  any  tw« 
sums  of  diverse  denominations. 

RULE  FOR  COMPOUND  SUBTRACTION. 

"Place  those  numbers  under  each  other,  which  arc  of  the  same  denomi- 
<'-.nation,  the  less  bc'un^  below  thegreatei';  begin  vith  the  least  dcnomina- 
"  nation,  and  if  il  exceed  tlie  figure  over  it,  borrow  as  many  units  as  make  one 
«  of  the  next  greater  ;  subtract  it  therefrom  ;  and  to  the  difference  add  the 
"  upper  figure,  remembering,  always  to  add  one  to  the  next  superior  denomi- 
"  nation,  for  that  which  you  borrowed. 

Proof.    In  the  same  manner  as  simple  Subtraction. 

1.  OF  MONEY. 

1.  Supposing  a  man  to  have  Ient;6*l85  10s.  7d.  and  to  have  rccerved 
again  of  his  money,  ^93   1 5s.  how  much  reraaisns  due  ? 


I.ent 
Received 

OPERATION. 
1. 

18    5         1    0, 
9    3        15 

d. 

7 
0 

From 
Take 

2. 

5   1   0 
8   5 

9. 

1    5 

Due 

9    1 

1    5 

7 

Zf7[ 

00 

Proof 

1    8   5 

1   % 

7 

3. 


Lent        6  3  7  1 

7 

8 

f     1   6  3 

Received  |          7  8 

at      <J          19 

Sundry  )       13  9 

limes.      13  2  6   1 

2 

4 

1    5 

6 

1 

5 
4 

Thb  sum  of  the  several 

payments  must  first  be  ad- 
ded togetlicr,  and  the  a- 
mount  subliacicd  from  the 
«um  lent. 


S4 


COMPOUND   SUBTRACTION.      Sect,  1.  6. 


4.  A  certain  man    sold  a  lot  of  land  for  £735   11    6  ;  he  received  at  on« 
time  jC61   5  :  at  another  time,  >C'195   13   U    how  much  is  there  yet  due  ? 
^  ,  4ns.  £i7^    12  7. 

e^r  21^   //  i_ 


^^ 


>t-cl.i 


^78       /Z     7 


2.  OF  TROY  WEIGHt. 


From     7     6 
Take            3 

1. 

or. 
8 
9 

inn. 
1      6 
I      7 

5T. 

1   3 
6 

3. 

lb.                OT. 

7               3 
2               8 

fiwt. 

5 
9 

Remain*  ^ 

10 

/^ 

-I'r 

"  4    6 

/^ 

Proof  f  ^ 

S 

/6 

!■& 

r'±. 

r 

5.  OF  AVOIRDUPOIS  WEIGHT. 


■f- 

1. 

o 

/^. 

cz. 

dr. 

T. 

cwt. 

gr. 

lb. 

9 

1      5 

5 

6 

1    1 

1 

1 

5 

•  « 

7 

1 

5 

I 

1 

dr. 
3 
8 


4     8/^      S    S    3  Z^  /<)  II 

9    /r    r 


Y.                    7)19. 

S      9^    .        6    . 
I      6*    ^        9 

1 

4.  OF  TIME. 

1. 
d.              h. 
6               2     0 
2               18 

•  4     4               5    45 
^5      9       A       5      Z    v 

fZ       ? 

X 

^         / 

^^      ^^ 

SfiCT.  I.  6.         COMPOUND  SUBTRACTION.  55 

5.  OF  MOTION. 
I.  2. 


• 

1     6 
8 

2    r 

3     4 

ft 

S      3 
2     3 

• 

6               8 
3               9 

5     1 

5   r 

7 

53 

■""27  ■ 

'  35 

2  ^^' 
6"    (5* 

--JT 

OF  CLOTH  MEASURE. 

1.  2. 

^rf».  yn  w.  -E.£.  qr.  Hi 

2     7  12  2      6  2  1 

16  13  17  3  2 


7.  OF  LONG  MEASURE. 

^^S'  mi.  fur.  p.  yds.  ft,  in.         bar, 

^6  13  526  2  1  8  1 

^     '^  15  2  27  1  2  9  2 

Ti         A3       J      26       Z~7  ^/ 


f .  OF  LAND  OR  SQUARE  MEASURE. 

1.  2. 

A.          R.        fiol.  fiol.                fi.  in 

17          117  is                 16  11 

16          1          16  10  201  130 


/  0     I        J__l6±_  Z£ 


5^             COMPOUND  SUBTRACTION.  Sect.  1  6. 

9.  OF  SOLID  MEASURE. 

1.  2. 

Tons.           ft.                 in.                      Cords.              ft.  in. 

45          2'    9                 186              68                 23  810 

1934          1237                     6          127  1529 

50ZZ1ZZ    IlZZMIUIl 


10.  OF  WINE  MEASURE, 

hhd. 

6      6 

1      7 

I                                            Tl  <  .V 

sal.            gfs.                       Uhd. 
3      1               2                         7      5 
3      3              3                         2     4 

1 

gal. 
1      6 
4      3 

^^  "gg 3  ■    7g  3 3^ 


II.  OF  ALE  AND  BEER  MEASURE. 


1. 

2. 

Hhd. 

sal. 

§r/*. 

Butt. 

hhd. 

.^a/. 

8     9 

1      9 

2 

6      S 

1 

1      6 

3     7  25  S  29  1  1^" 


S}     ^r    3  34     0       I 


12.  OF  DRY  MEASURE. 

I.  2. 

^"-  y^-^-.  ?^9.  CkaL  bu.  Jik. 

6      1  1  2  1      7      I  I      3  1 

5  14  7     6  2     3  2 


^S   S    6     •   __.f7^  '17   3 


THE 


SCHOLAR^S  ARITHMETIC 


OBSERVATIONS. 


HE  Scholar  has  now  surveyed  the  ground  work  of  Arithme- 
tic. It  has  before  been  intimated,  that  the  only  way  in  which 
numbers  can  be  affected  is  by  the  operations  of  Addition,  Sub- 
traction, Multiplication  and  Division,  These  rules  hav^  now  been 
taught  him,  and  the  exercises  in  a  supplement  to  each,  suggest 
their  use  and  application  to  the  purposes  and  concerns  of  life. 
Further,  the  thing  needful,  and  that  Which  distinguishes  the  A- 
rithmetician,  is  to  know  how  to  proceed  by  application  of  rZ^d-i-^? 
four  rules  to  the  solution  of  any  arithmetical  question.  To  af- 
ford the  scholar  this  knowledge  is  the  object  of  all  succeeding 
rules. 

^  -::<!  4:c-  -r.:-  i(.  -;.:•  >::<  -;:?■  —       — 

SECTION  II. 


Rules  essentially  necessary  for  cucry  person  to  fit  and  qualify  them 
for  the  transaction  of  business, 

Tkese  arc  nine  :  reduction,  fractions,*  federal  money,  interest^ 

COMPOUND  MULTIPLICATION,  compound  DIVISION,  SINGLE  RULE  OF  THREE) 
DOUDLE    RULE    OF    THREE    and  PRACTICE. 

A  THORouoH  knowledge  of  hiese  rules  is  sufficient  for  every  ordinary  oc- 
curntnce  in  life.  Short  of  this  a  person  in  any  kind  of  business,  will  be  liable 
to  repeated  embarrassments.  It  is  the  extreme  usefulness  of  these  rules 
which  commends  them  to  the  altenUQU  of  every  Scholar. 

*  Fr  ACTIOS  s  are  tairn  u/i  here  no  further  than  is  necessary  to  shew  their  tiff" 
nification^aihd  toilluutratc  the /irinci/ilci  of  FMOMJiAi  JMoNgr, 

H 


SS  REDUCTION.  Saci.  II. 

§  1.  i^^tjuction- 


^j^-v::-4::-v^< 


"  RiJDt'CTioN  teaches  to  bring  or  exchange  numbers  of  one  denommalion 
t'  to  otheis  of  different  denominations,  retaining  the  same  value." 

//  is  of  two  kinds, 

I .  When  high  denominations  are  to  be  brought  into  lower,  as  pounds  into 
shillings,  pence,  and  fartJiings  ;  it  is  then  caiicd  reductio-n  descending-, 
and  is  ptrformed  by  MulUjilication. 

II.  IVhen  lower  deriominations  are  to  be  brought  into  higher,  as  farthings 
into  pence,  or  into  pence,  •  shillings  and  pounds  ;  it  is  then  called  reduc- 
tion ASCENDING,  and  is  performed  by  Division. 


Reduction  Descending, 


RULE. 

Multiply  tlie  highest  denomination  by  that  number  which  it  takes  of  th» 
next  It  SS  to  make  one  of  tirat  greater  ;  so  continue  to  do,  till  you  have  brought 
it  as  low  as  your  question  requires. 

Pkoof.  *'  Change  the  order  of  the  question,  and  divide  yonj?  last  product 
by  the  labt  multiplier,  and  so  on.'*i 

E2?AMPLES. 

1. 1^  £\7  13s.  6rf.  "qrs.  how  many  farthings  ? 

orERATTON. 

£.        s.     d.     g.rs.  In  this  example,  the  highest,  denonl - 

17     13     6       3  tnalion  is  pounds,  the  next  less,  is  shil- 

2  0  Shillings  inpoimd.  Hngs,   and  because  20  shillings  make 

— one  pound,  therefore,  I  multiply  ^'17  by 

3  5   3    Shillings  In  £\7    13s.  20,  increasing  the  product  by  the  addi- 
l   2  Faice  in  a  shilling,        tion  of  the  given  shillings,  (13)  which 

it  must  be  remembered,  must  always  be 

4242  Fence  ill  jQ\7  ISe^.  6c?.  done  in  like  cases  ;  then,  because  13 
4  Farthings  znaj2enny.     pence  make  one  shilling,  J  multiply  the 

.^ shi-Iings,  (353)  by  12,  adding  in  the  §^iv- 

./T71SA  &  9  7   1       Farthings  en  pence  (6f/.)  lastly  because  4  farthings 

make  one  penny,!  multiply  the  pence 
(4242)  by  4  and  add  in  the  given  farthings  (Syr^.)  I   then  find,  that, in    17 
'35.  ed.  b(jrs.  there  are  16971  fratbings.  '^■ 

PROOF. 

4)  1   6  9  7  1  To  prove  the  above  question  change  the  ordes^ 

— —  of  it,  and  it  will  stand  thus  j  in  16971  farthings, 

12)4  2  4  2     3_:^rs.     hov/  many  pounds  ? 

Divide  the  last  product  by  the  last  multiplier, 

2jO)  3  DiS     6d.         the  remainder  will  be  fart.'iings.     Proceed  in  tlus 

— v/ay  till  iiii  the  steps  oC  the  operation  have  been 

£1   7       IZs.  retraced  back  ;  the  last  qiftotient  with  the  remain- 

ders will  be  proof  of  the  accuracy  of  tlie  opera- 
tion if  the  j?  agree  ^\ilh  tlae  sum  given  in  the  ques- 
lioG. 


I 
Sect.  If.  L  REDUCTION.,  69 

2.  l^  £7  \A5.6d.   I^r.  how  marif        .3.  In  ^7  6s.  Ad,  how  many  pence  ? 

Earthings -f*  J?}».  7417  ^rs,  ^  Ans,   \7o^d. 

7./^.6//  ■  7.6.4 


2<'o 


4.  In  29  Guineas,  at  28s- how  many       5.  In ^173  15?.  how  many  sixpen- 
farthln^^s        ^««.  3§9  76  ^s.  ces  ?  ^^jy.  69oO. 


Y.  r.>r  671  eai>Ies,   at  10  dollars,  each, 
how  mmy  shillings,  threepences,  pence 
6.  In  12  crowns,  at  Cf7^  how       and  favlliiii^-s  ?  Jwi.  A02Zd  ff-.iU. 

liny  pence  and  lunliinj^s  ?  1G1040  three  fienccsy  4^^120  /.  i 

Ann.  94J3  d.  37^2  (jn-         193?4ill>  fjrs^ 

6    /  6 


"TWa —  ^ 


60 


REDUCTION. 


Sect.  II.  .1 


Reduction  Ascending. 


RULE 


Divide  the  lowest  denomination  given  by  that  number  which  it  takes  of 
the  same  to  make  one  of  the  next  higher,  and  so  continue  to  do,  till  you  have 
brought  it  into  that  denomination  which  your  question  requires. 


EXAMPLES 
I.  In  16971  farthings  how  many  pounds  ? 

OPERATION. 

Farthings  in  a  ^lenny  4)16971 
Pence  in  a  s/iiiHng      12)4242   Sqrs. 
JShiilings  in  a/iound     2/0)35)3  6d. 

j^ns.  17   136(.  6d.  Sgrs, 


■■%: 

Reduction  descdfifiing  and  as- 
cending reciprocstSBbrove  each 
other.  ^ 


2.  In  1765  pence,  how  many 
pounds  ?  jins.  £7  7s.   \d. 


4.  In   38976  farthings,  how  many 
guineas  ?         Ans.  29. 


t9  ^j^- 


4.  In  6950  sixpences,  how  many 
pounds  ?  Ans.£\7o.  I5s 


5.  In  3792  farthings,  how  many 


crowns 


Ans,      12. 


I S8 


Sect.  II.  1, 


REDUCTION. 


61 


6.  In  48960  farthings,  how  many  pence,         7.  In  6952  three  pences,  how 

three-pences,  six  pences,  and  dollars.  many  pistoles  at  22s.  each  ? 
jins»  l'2'24=0  pence,  4080  i/irec'/icnces,  jins.79, 

2040  six'pences.     170  chllars^ 


6)  1020    ^^'V^' 

lyo       . 


/9  8 


Reduction  Ascending  and  Descending. 


1.  BIONEY. 


1.  In  57  nioidores,  at  36s.  each, 
how  many  dollars  ? 

Ans.  342  dollar: 


3  3^Z 


In  this  question,  the  first  s;tep  will  be 
to  bring  the  moidores  inco  shillings  I 
lastly  bring  the  shillings  into  dollars. 

2.  In  75  pistoles  ho\v  many  pounds  ? 
Ant.  £82  lOs. 


3.  In  ^73  how  many  guineas. 
uiJis.  52  ^tiinea/if  4*. 


Z0 


/40    V 


60 
56 


>/^//. 


4.  In  ^63  and   5  guineas  how 
many  dollars  ? 

Am.  233  dollart.  2.5. 


— 1^' 


^L 


63 

~nnT?r 


^ 


62  REDUCTION.  Si:ct.  II.  1, 

"  JVhen  it  is  required  to  hioiv  hoto  many  sorts  of  coin  of  different  values^  and 
of  equal  number^  are  contained  in  amj  number  of  another  kind  /  reduce  the  seVf 
cral  sorts  of  coin  into  the  lowest  denomination  mentioned,  and  add  them  to- 
gether for  a  divisor  ;  then  reduce  the  money  given,  into  the  same  denomintv- 
tion  for  a  dividend,  and  the  quotient  arisiinj  from  the  division  will  be  the 
number  required." 

<f  J^^e^E,  Observe  the  same  direction  in  weights  and  measures." 

1.  In  «I4  guineas,  how  naany  pouads,  dollars  and  shillings  of  each  an  equal 
fi amber  ? 

OPERATION. 

1  is  20  shilling*  54  guineas 

A  dollar  is  6  shillings  28  Shilling  is  a  guinea 

1   shilling  

—  432 

Vivisor.  27   Shillings  108 

Vividefidy  1512  shillings. 

37)1512(56  of  each  ;  that  js,  54  guineas  include  the  vajue  of  ope  pound, 
135  ojie  doUar,  and  one  shilling,  56  times. 

162 
162 

000 

2,  I>-  172  moidores,  how  many  eagles,  dollars  and  nine-pences,  of  each  the 
like  Ttumber  ?  Jns,  92  of  each ^  and  68  nine-^iences  over, 

56^. 


/0S2. 

rzo9  • 


Ecr. 

IL 

1. 

REDUCTION. 

TROY  WEIGHT, 

1.  In  Aid. 

5oz, 

\6Jiwis.  how  many  grains  ? 

lb. 
4 
12 

OPERATION. 

oz.    /iwts. 
5           16 
oz.  in  a  fiound^ 

53 
20 

Ounces, 
fiivts.  in  an  ounce. 

1076 
24 

Penny  weights, 
grs.  in  one  fiivt. 

Fr 

00/, 

4364 
2152 

24)25824 
20)1075 

(xrainsy  the  answef^ 
16  /iwts. 

12)53 

5oz.. 

4 

lb. 

C3 


%.  Ik  \Olb.  of  silver,  how  many  spoons,  each  weighing  5  az*  \0  fit^Si 
Amt'%1  s/ioontt)  and  9Q  fiwJs,  over, 

/.^  »t  If) 

2.0  120 

I  I  Out:  /  20     • 

^  ik>)  14  olo      , 


..  A 


64  REDUCTION.  Sect.  IL  1 

5.  In  45681  grains  of  silver  how  many  pounds? 

OPERATION. 

20  12 

34)45.68T(1903('95(7/^>,     Jnsiver,  7}h .     l\oz,  Spivts.  9grs, 
24  180     84  12 

216  103    11  oz.  95 

21S  100  20 

081      003 /iivts»  1903 

72  24 


09  grs,  45681  Proof. 

4.  In  4560  grains  of  silver,  how  many  tea-spoons,  each  one  ounce  ? 
Jns.  9^tea-s/2007i8. 


2J  6 


0 


3.  AVOIRDUPOIS  WEIGHT. 

Civt.  qr.  lb.    oz. 
I.In  er     1     13     11   how  many  drams  ? 


269 
28 


2165 

538 

7545 
16 

452*1 

7545 

120731 
16 

724386 
120731 

1931S96 


PllOOF. 

16)1931696 

16)120731 

11    02. 

28)7546 
4)269 

13  lb. 
1   qr. 

«7 

/ 

Civt. 

Sect.  II.  1.  REDUCTION.  65 

2.  In  14048  oz.  how  many  hundred  weight  ?         Ana.  7C.  Sqrs.  10/6. 
2ff  ifri'^       O^i.  9*-    ^ 


J  \  J)  /I  '    '  'a/j  <;a> 


3.  In  470  boxes  of  Sugar,  each  26/6.  how  manyCwt.  ?  Ans.  109  C.  oqra.  \2lb 

2810' fs 

Ton 

8U 


m. 


4.  In  17Cwt.  Iqr.  6lb.of  Sugar,  how  many  parcels,  each  \7  lb.? 

t/J.6     . 
4l 

275 

_L2L 

€<9 


66 


i                             REDUCTION. 

Sect. 

4  TIME. 

1.  In  121812  seconds,  how  many  hours! 

OPERATION. 

pRooy. 

6)0)121812 
6lO)203[0      12  sec. 

H.     m.     s. 
33      50      12 
60 

Ans.  33A.  SOm.   12s. 

2030 
60 

\ 


121812 
2.  Supposing  a  man  to  be  21  years  old,  how  many  seconds  has  he  lived, 
allowing  365  days,  6  hours  to  a  year.  Ans.  66^2709600  seconds. 

750 

d  7S6 

60 


sTsTtToc 

tTzToJJc7  c^47^  - 

5.  How  many  minutes  from  the  commencement  of  the  war  between  A- 
merica  and  England,  April  19,1775,10  the  settlement  of  a  general  peace 
whichtookplace,  Jan.  20,  1783  ?  Ans.  ^^7^  \^0  minutes. 


^  _I&1. 


^rti^^ 


T¥J6 


wv. 


^0  7  9 1' 6  a 


Sect.  II.  1.  REDUCTION. 

4.  In  413280  nJnutcs  liow  many  weeks?  Jins.  41. 


^8 

IG8 
/6(9 


67 


5.  LONG  MEASURE. 


1.  Reducji  16  miles  to  barley-c9rns. 

OPERATION. 

1,6  Miles. 
8 

128  Furlongs. 
40 


5120  R9d8. 


2560Cr 
2560 

28160    Yards. 
3 


1>ROOF. 

3)3041280 

12)1013760 

3)844SX) 

t    11)28160 

2560 

2 

4|0)512[O 
8)128 


84480  Feet 
12 


1013760  Inches. 
3 


16  Miles. 

t  Divide  by  1 1  for  5^  and  multipljr 
the  quotient  by  2.  The  reason  is  be- 
cause 5  }  reduced  to  half  yards  is   11. 


Ans.  2Q^\2tQ  Barley  Corns. 
*To  multiply  by  one  half{^)  it  is  only  to  take  half  the  Dividend. 
2.  In  47530  feet  how  many  Leagues  ?  Jins.  3  Leagues. 

Z 


4i 


5 


68  REDUCTION.  Sect.  II.   1. 

.1.  How  many  times  does  the  Wheel,  which  is  18  feet  6  inches  In  circuTn- 
ference,  turn  round  in  the  distance  of  150  miles? 

'U-    /*^  .^^5.  42810  ;/mf5,  anci  180  znc/zM  f9rv-r„ 

—^-4^       /  /20C 

1»  // 

/_%_     /^  --       ,/ 

S  (9  (9     *  *  *    ^ 


f  900 


2^0 

4.  How  many  barley-corns   will  reach  round    the  Globe  it  being  360  dc: 
grees  i  «fns.  4755801600 


.i^^ 

t 


G9-^ 


2.1  BO 

I  so 
8 


ZCo I  60 

2/^05  rZ  60 
3 

/2. 

3 

1/TsTsoi  boo  u4ii. 


Sect.  II.  1.  REDUCTION. 

6.  LAND  OR  SQUARE  MEASURE. 
^.  In  13  acres,  2  roods'^iow  many  poles  ? 


^ 


OPERATION. 

Jc.     r. 
13      2 


S4> 
40 


PROOF. 

4'0)2l6iO 


4)54 

13  ^c.  2  roodi. 


Ans.  2  1 60  Poles 
5.  In  2852  rods  how  many  acres  ? 


Ans,   \7  A,  3  R.  \2  P., 


9.  SOLID  MEASURE. 
1.  In  1296000  solid  inches,  how  many  tons  of  hewn  timber  ? 


oreration; 

5,0 
1728)1296000(7510 
12096     

the  Answer 

rRoor- 
15 
50 

750 
1728 

6000 
1500 
5250 
750 

8640 
8640 

00 

1396000  I/u-/ic*. 

70 


REDUCTION. 


2.  Ix  552960G  solid  inches,  how  many  cords  of  wood  ? 


Sect.  II: 

Jns,  25 


3^M 
oo 


8.  DRY  MEASURE. 
1  In  75  bushele  of  corn  how  many  pints  ? 


OPERATION. 

4 

300 
8 


2400 
2 


PROOF. 

2)4800 

8)2400 
4)300 


75  bushels. 


^ns.    4800  pints 
2.  In  9376  quarts  how  many  bushels?         jins.  293. 

zss 


It  would  be  needless  to  ^ive  examples  of  Reduction  in  all  the  weights  and 
measures.  The  understanding,  which  the  attentive  Scholar  must  already  have 
acquired  of  this  rule,  by  help  of  the  Tables,  will  e-ver  be  sufficient  for  his  pur- 
pose. 


Sect.  I.  II.      SUPPLEMENT  to  REDUCTION. 


Supplement  to  IHctlUCtiOn* 

mm'  -;>  -r  ■!•>  -r  v.c-  -^  >^  -^  -J'.c-  ^  .-J.  wT  4;;-  — 

QUESTIONS. 

1.  WHAf  is  Reducticn. 

2.  O/-  Aow  7nany  kinds  is  Reduction  ?  nvhat  are  they  called  ?  toherein  do  these 
kinds  differ  one  from  the  other  ?  Which  of  these  fundamental  r\des  are  em* 
ployed  in  their  operations  ? 

0.  How  is  Reduction  Descending  performed  ? 
A.  Hoiv is  Reduction  Ascending  perjormed? 
5.   When  it  is  required  to  know  /tow  many  sorts  of  coin ^  nveights  or  measures  of 

different  values^  of  each  an  equal  number ^  are  contained  in  any  other  number 

of  another  kindy  wMt  is  the  method  of  procedure  ? 

EXERCISES. 
1.  In  36  guineas  how  many  cEowns  ?     Jns,  153  croions,  and  9d.  over. 


IOCS' 

~TiT 

2Z/6 

r 


12  SUPPLEMENT  to  REDUCTION.       Sect.  II.  1. 

2.  How   many  steps  of  2  feet  5   inches   each,  will  it   require   a  man  to 
take,  going  from  Leominster  to   Boston,  it  being  43  miles. 

i/ins.  93948  ste/is ;  JCI/**M<?  last  steji  mil  carry  him  into  the  toivn  12  inches. 


3^^ 
4/) 

II 

z)  15-1  dec 

7S6  BO 

5 

1%7  OLI  0 

%6 1 

"A 

Z7^ 
Z6I 

f5^ 
ZC5 

/y^TyiA^ri^t*"   /  /^*^/;£^. 


,  5.  Let  70  dollars  be  distributed  among 
tliree  men  in  such  a  manner  that  as  often 
as  the  first  has  SAthe  second  shall  have  7J 
and  the  third  9/  What  will  each  one  re- 
ceive ?  jins.  Fir  at  1 6  dolls.  A>J  Second 
23  dolls.  2/  Third  30  dolU' 


^% 


6)1  0  0  ,       ,    , 


/^ 


6/700 


Sect.  II.  1.      SUPPLEMENT  to  REDUCTION.  73 

4.  If  a  vinter  be  desirous  to  draw  off  a  Pipe  fif  Canary  into  bottles  contain- 
ing pints,  quarts,  and  2  qiiurts,  of  each  an  equal  numbtr,  how  many  must 
he  have  ?  ^ns.  144  gJ  each. 


T)I0  08       , 


5.  There  are  three  fields  ;  one  contains  7  acres,  another  10  acres  and 
the  other  12  acres  and  1  rood  ;  how  many  shares  of  76  perches  each)  are 
contained  in  the  whole  ?         Ans.  61  shares  and  44  Jierc/ies  ever. 


t    1^ 

7 

f^     '. 

/z-/ 

£9-/ 

4^ 

/  /^/^      o-^^-rh 

^^ 

F6)  46  80  [6 J.  ^^w^'^^. 

l%0 

r^ 

f 

/ 

74  SUPPLEMENT  TO  REDUCTION.      Sect.  II.   1. 

6.  There  are  1061b.  of  silver,  the  property  of  3  men  ;  of  which  A  receives 
\7lb.  XOnz.  \9/iiv(fi.  I9grs.  of  what  remains,  B  shares  lor.  7grs.  so  often  as  C 
shares  \3fiwts.  W^at  are  the  shares  of  B.  and  C  ? 

Ansnver,  B*s  share  5ilb.  ioz.  5/iwts,  S^rs,     C*s  share  24lb.  4oz  AB/i^ts, 

/G6  .  00.  00.   00         ,      . 
I />"  .    /  O  '  19  .  /?    ^  <^X^/^ 

ii^  to  /y 

^^^^ 


7i*rt5K'*^^ 


/to 


I  f  f.^. 


Sect.  II.  2.  FR ACTONS.  75 

§  2.  fraction,^* 

—      !■■  "5:>  -r  -,::-  ^  -i'.j-  -r  j^c  -/■  -;.s-  ^  -v.*  ->~  -/.;-  •■        i 

When  the  thing  or  things  signified  by  figures  are  ivhoh  ones,  then  the  fig- 
ures which  signify  them  are  called  Integers  qv  ivhole  numbers.  But  ui.eii 
only  some  parts  of  a  thing  are  signified  by  figures,  as  tivo  thirds  of  any  thing, 
Jive  sixths,  seven  tenths^  Ij'c.  then  the  figures  which  signify  these  f.arts  of  a 
thing  being  the  expression  of  some  quantity  less  than  owe,  are  called  Frac- 
tions. 

Fractions  are  of  two  kinds,  Vulgar  and  Dedmal ;  ihey  are  distinguished 
by  the  manner  of  representing  them  ;  they  also  differ  in  their  modes  of  ©pera- 
tion.  \ 

VULGAR  FRACTIONS. 

To  understand  Vulgar  Fractions,  the  learner  must  suppose  an  integer  (or 
the  number  1)  divided  into  a  number  of  equal  parts  ;  then  any  number  of 
these  parts  being  taken,  would  make  a  fraction,  which  would  be  represented 
by  two  numbers  placed  one  directly  over  the  other,  with  a  short  line  between 
them,  thus  -|  two  thirds,  |  three  fifths,  ^  seven  eights,  ^c. 

Each  of  these  figures  have  a  different  name  and  a  different  signification.  The 
figure  below  the  line  is  called  the  Denominator  and  shews  into  how  many 
parts  an  integer,  or  one  individual  of  any  thing  is  divided— the  figure  above 
the  line  is  called  the  nwnlerator  and  shews  how  many  of  those  parts  are  sig- 
nified by  the  fraction. 

For  illustration,  suppose  a  silver  plate  to  be  divided  into  vine  equal  fiarts. 
Now  one  or  more  ot  these  parts  make  a  fraclien,  which  will  be  represented  by 
the  figure  9  for  a  denominator  placed  underneath  a  short  line  shewing  the  plate 
to  be  divided  into  nine  equal  fiarts  y  and  supposing  I'-iUo  of  those  parts  to  be 
taken  for  the  fraction,  then  the  figure  2  must  be  placed  directly  above  the  9 
and  over  the  line  (-Dfor  a  Numerator,  shewing  thai  ivv©  of  those  parts  are  sig- 
nified by  the  fraction,  or  t%vo  ninths  of  the  plate.  Now  let  5  parts  of  this  plate, 
^hich  is  divided  into  9  parts,  be  given  to  John  his  fraction  would  be  ^five 
ninths  ;  let  3  other  parts  be  given  to  Harry,  his  fraction  would  be  -i  three 
ninths;  there  would  then  be  one  part  of  the  plate  remaining  still  (5  and  3  arc  8) 
and  this  fraction  would  be  expressed  thus  -i  one  ninth. 

In  tliis  way  all  vulgar  fractions  are  written  ;  the  Denominator,  ornumber 
below  the  line  shewing  into  how  many  parts  any  thing  is  divided,  and  the  nuw 
merator,  or  number  above  the  line,  shewing  how  many  of  those  parts  are  ta- 
ken, or  signified  Uy  the  fraction. 

To  ascertain  whether  the  Learner  understands  what  has  now  been  taught 
kim  of  fractions,  let  uf  again  suppose  a  dollar  to  be  cut  into  13  equal  parts;  — 
let  2  of  those  ])aris  be  given  to  A  ;  4  to  B  ;  and  7  to  C. 

f  A's  fraction .   -^ 

Required  of  the  Learner  that  he  should  wiite<^  B's  fraction .  J^ 

LC's  fraction .  -f^ 

It  is  from  Division  only  that  fractions  arise  in  Arithmetical  operations  : 
the  remainder  after  division  is  a  portion  of  the  Dividend  undivided;  and  is 
always  the  Numerator  to  a  fraction  of  which  the  Divisor  is  the  Denomina- 
tor.    The  Quotient  is  so  many  integers. 

The  Arithmetic  of  Vulgarl'ractions  is  tedious  and  even  intricate  to  beginners. 
Besides,  tlicy  are  not  of  necessary  use.  We  shall  not,  therefore,  enter  into  any 
further  coniiideration  of  lliem  here.  This  difficulty  arises  chitfiy  from  the  varie- 
ty of  denominators  ;  for  when  numbers  are  divided  into  diffvrent  kii-ds,  or  parts 


76  DECIMAL  FRACTIONS.         Sect.  II.  2. 

they  cannot  be  easily  compared.     This  consideration  gave  rise  to  the  inven- 
tion of 

DECIMAL  FRACTIONS. 

Decimal  Fractions  are  also  expressions  of  parts  of  an  integer  ;  or,  are  in 
value  something  less  than  one  of  any  thing,  whatever  it  may  be,  which  is  sig- 
nified by  them. 

In  decimals,  an  integer,  or  the  number  one,  as  1  foot,  1  dollar,  1  year,  Sec. 
is  conceived  to  be  divided  into  ten  equal  parts,  (in  vulgar  fraaions  an  integer 
may  be  divided  into  any  number  of  parts)  and  each  of  these  parts  is  subdivided 
into  ten  lesser  parts,  and  so  c%.  In  this  way,  the  denominator  to  a  decimal 
fraction  in  all  cases,  will  be  either  10,  100,  1000,  or  unity  (1)  with  a  number 
of  cyphers  annexed  ;  and  this  number  of  cyphers  will  always  be  equal  to  the 
number  of  places  in  the  numer%M^  Thus, -j^^ -^y^  ^^^^^  are  Z^eczma/ i'Vac- 
/fo?2.<j,  of  which  the  cyphers  in  the  denominator  of  each  are  equal  to  the  num- 
ber of  places  in  its  own  numerator. 

*' As  the  denominator  of  a  decimal  fraction  is  always  10,  100,  1000,  Sec. 
*•*  the  denominators  need  not  be  expressed  ;  fV  r  the  numerator  only  may  be 
"  made  to  express  the  true  value  ;  for  this  purpose  it  is  only  required  to 
"  write  the  numerator  with  a  point  (  ,  )  before  it,  called  a  separatrix^  at  the 
"  left  hand  to  distinguish  it  from  a  whole  number  ;  thus,^^^  is  written  ,6  ;  -f^-^ 
^27  J  tWo  »6S5,  &c. 

When  integers  and  decinaalsare  express  d  together  in  the  same  sum,  that 
sum  is  called  a  vuxcd  number  ;  Thus,  25,63  is  a  mixed  number  ;  25,  or  all 
the  figures  to  the  left  hand  of  the  separatrix  bting  integers,  and  ,63  or  all 
the  figures  to  the  right  hand  of  the  same  point  being  decimals. 

The  first  figure  on  the  right  hand  of  the  decimal  point  signifies  tenth  parts, 
the  next  hundredth  parts,  the  next  thousandth  parts,  and  so  on. 

,7  signifies  seven  tenth  parts. 

,07 seven  hundredth  pans. 

,27 two  tenth   parts  and  seven  hundredth  parts  ;  or  twenty-seven 

hundredths. 

,357 three  tenth  parts,  five  hundredth  parts,  and  seven  thousandth 

parts  ;  or,  357  tliousandths. 
5,7          five  and  seven  tenth  parts. 
5,007  -five  and  seven  thousandths. 

The  value  of  each  figure  from  unity,  and  the  decrease  of  decimals  toward 
the  right  hand,  may  be  seen  in  the  following 

TABLE. 


^    t:    -a   -u    cu  ^    ^ 
■Z    "3     c    a     r-    ^    -z 

rt      ^     2     c     o     o 


lA     ffi     (/I     a     c    ""^     ^  rt 

.2   .2   .2    S    S    ^    ^  ,     ^    ^    is    I    g    c   -^   - 

--  -^        -  £  J    g   g  -H   S    §  -§  H  H  5  ^  fe 

5  43212,  3456789 

Cyphers   placed  to  the  right  hand  of  decimals  do  hot  alter  their  value, 
placed  at  'the  left  hand,  they  diminish  their  value  in  a  tenfold  proportion. 


^ 

S 

^ 

H 

o 

X 

O 

9 

8 

7 

6 

Sect.  II.  2.         DECIMAL  FRACTIONS. 


/  / 


ADDITION  OF  DECIMALS. 


RULE. 


"1.  Place  the  numbers  whether  mixed  or  pure  decimals,  under  each 
other,  accordinpj  to  the  value  of  their  places.'* 

"2.  Find  their  sum  as  in  whole  numbers,  and  point  off  so  many  places 
for  decimals  as  are  equal  to  the  greatest  number  of  decimal  places  in  any  of 
the  given  numbers.'* 

EXAMPLES. 

1 .  What  is  the  amount  of  73,5 1 2  gwineas,  43  6  guineas,  3,27  guineas,  ,8632 
of  a  guinea,  and  100,19  guineas,  when  added  together  ? 

OPERATIQN. 

The  decimals  are  arranged  f»'om  the 
separatrix  towards  the  right  hand  and 
the  whole  numbers  from  the  same  point 
towards  the  left  hand.  The  greatest 
number  of  decimal  places  in  any  of  the 
numbers  is  four,  consequently,  four  fig- 
ures in  the  product  must  be  poinled  off 
for  decimals. 


73,612 
436, 
3,27 
,8632 
100,19 

Ans.  613,9352  ^-uineas. 


2. 
343,601 
,3724 
63,1 
572,813 
7,5462 


3.    Required   the    sum  of  37,821-}- 
546,35  +  8,44-37,325  ? 

Antwer,  629,896, 

,896  M. 


Tzf 


4.  What  is  the  sum  of  three  hundred 
twenty  nine  and  seven  tenth  ;  thirty  seven 
and  one  hundred  sixty  two  thousandths 
und  sixteen  hundredths  when  added  to- 
gether. Ans.  367,022. 


5i>T  ,OZZ 


5.  Add  six  hundred  and  five  ihaii- 
sandths,and  four  thousandrfc*  and 
three  hundredths  \  Suniy  4600,035 

'  '2/1  do; OSS'' 


Note.  When  the  numerator  has  not  so  many  places  as  the  dononiin.^- 
tor  has  cyphers,  prefix  so  many  cyphers  at  the  left  l.anil  ns  nvIII  uvxkv  va  tlie 
defect ;  so  y^',^  is  written  thus,  ,005,  &c. 


78 


DECIMAL  FRACTIONS. 


Sect.  II.  2. 


SUBTRACTION  OF  DECIMALS. 


RULE. 


«  Place  the  numbers  according  to  their  va'ue  ;  then  subtract  as  in  whole 
numbers,  and  point  oif  the  decimals  as  in  addition.'* 


EXAMPLES. 


1.  From  716,325  take  81,6201. 

OPERATION. 

i^ro7«  716,325  ^ 

Take     8  1,6201 


Fern.  634,7049 


2.  From  119,1384  take  95,91. 

Bern.  33,2284. 


3.  What  is  the  difference  between 
2S7  and  3,1 15  ?  Answer^  283,885 


28d;S8S 


4.  From  67  <.ake  ,92 
Bern.  66,08 


b&  f  0  8 


All  the  operations  in  Decimal  Fractions  are  extremely  easy  ;  t^he  only  li- 
abiHly  to  error  will  be  in  placing  the  numbers  and  pointing  ofFthe  decimals  ; 
and  here  care  will  always  be  security  against  mistakes. 


MULTIPLICATION  OF  DECIMALS. 

RULE. 

"Whether  they  be  mixed  numbers,  or  pure  decimals,  place  the  factors 
and  multiply  them  as  iu  whole  numbers." 

"  2.  Point  off  so  many  figures  from  the  product  as  there  are  decimal  pla- 
ces in  both  the  factors  ;  and  if  there  be  not  so  many  decimal  places  in  the 
product,  supply  the  detect  by  prefixing  cyphers.'* 


EXAMPLES. 


1.  Multiply,  ,0261  by  ,0035 

OPERATION. 

,0261 
,0035 


1  305 


In  this  example,  the  decimals  in  the 
two  factors  taken  together  are  tij^ht  ; 
the  product  falls  short  of  tliis  number  by 
four  figures,  consequently,  four  cyphers 
are  prefixed  to  the  left  hand  of  the  pro- 
duct. 


5O00OL)135//rou'i/r;. 


Sect.  II.  2. 


DECIMAL  FRACTIONS. 


79 


2.  Multiply  31,72  by  65,3 
Product y  2071,316 


operation; 
S  1,  r  2 
6  5,3 

9^1  s 

IS860 

1671,5  I  6 

4>.  Multiply  ,62  by  ,04, 
Product  ,024^ 

,6% 


.[0%^S 


5.  Multiply  25,238  by  12, ir 
J'roduct,  S07,li6i6 

2^-13  8 

S.  Multfply  17,6  by  ,75 
Producty  13,3 

0  80 
Id'}  Z 
/3,ZO0 


DIVISION  OF  DECIMALS- 


RULE. 

*'  1 .  The  places  of  decimal  parts  in  the  divisor  and  quotient  counted  togeth- 
er must  be  always  equal  to  those  in  the  dividend,  therefore  divide  as  in  whole 
numbers,  and  from  the  rit^ht  hand  of  the  quotient,  point  off  so  many  places 
lor  decimals,  as  the  decimal  places  in  the  dividend  exceed  those  in  the  divisor. 

"  2.  If  the  places  of  the  quotient  be  not  so  many  as  the  rule  requires,  sup- 
ply the  dtfcct  by  prefixing  cyphers  to  the  left  hand. 

"3.  If  at  any  time  there  be  a  remainder,  or  the  decimal  places  in  the  di- 
visor be  more  than  those  in  the  dividend,  cyphers  may  be  annexed  to  the  div- 
idend, or  to  the  remainder,  and  the  quotient  carried  on  to  any  degree  of  ex- 
actness." 


Divide  2,735  by  51,2 

OPERATION. 

51,2)2, 735(,0534-f 
2  560 


1750 
1536 


2  140 
2048 


EXAMPLES. 


In  this  example,  there  are  ^five  decimal*  in 
the  dividend  (counting  lUe  two  cyphers  which 
were  added  to  the  remainder  of  the  dividend 
after  the  first  division)  t!»at  the  decimals  in  the 
divisor  and  quotient  counted   together  may  c- 

qual  that  number,  a  cypher  is  prefixed  to  the 

left  hand  of  the  quotient. 


92 


80 


DECIMAL  FRACTIONS. 


Sect.  II.  2, 


Ik  division  of  decimals  it  is  proper  to  add  cyphers  so  lorg  as  there  contin- 
ues to  be  a  remainder,  this  however  is  not  practised  nor  is  it  necessary  ;  four, 
or  five  decimals  being  sufficiently  accurate  for  most  calculations. 


2.  Divide  3156,293  by  25,17. 
Quotient,  1253-f. 


The   Scholar  is  requested  to  point 
the  following   example  as  the   rule  di*^ 


rects. 


4.  Divide,  173948  by   ,375. 
^wo^^nr,  463861-j- 

nsc 


MOO 
-^500 


^00 


3.  Divide  5737  by  13,3 
Quotient ^  4313534- 


~m 


Divide  2  by  53,1 
Quotient ,  Qo7-\- 

mr 


Sect.  II.  2.  DECIMAL  FRACTIONS.  81 

REDUCTION  OF  DECIMALS. 
CASE  1. 
To  Reduce  Vulgar  Fractions  to  Decimals. 

RULE. 

Annex  a  cypher  to  the  numerator  and  divide  it  by  the  denominator,  annex- 
ing a  cypher  continually  to  the  remainder.  The  quotient  will  be  the  decimal 
required. 

EXAMPLES. 
1.  Reduce  4  to  a  decimal.  2.  Reduce  |  to  a  decimal. 

OPERATION.  OPERATION. 

5)3,OC,6  ./fniw^r.        The  numerator  in  these  7)I,0(,1428+^nff, 

3  0                         Operations  is  considered  as  7 

■                      an  integer,  and  always  re-  — 

0  0                          quires  the  decimal   point  30 

to   be  placed  immediately  28 

afte-r  it,  the  cyphers  annexed  occupy  the  places  of  — 

decimals,  the  quotient  must  be  pointed   off  ac-  20 

cording  to  the  rule  in  Division.  14 

60 
56 


5.  Reduce  i, -J,  and  ^  to  decimals.    Antwer9)  ,25.  ,5.  ,7S, 


^;a^(.2^ 

2)1^^   . 

m{u 

zo 

"^^ 

-^ 

M- 

%o^ 

TT 

ov 

4.  Reduce  ^^j,  ^^y 

andTiW  to  decimals.  An^  1923+j025  ,007974. 

ff2.9)y*ooa{,ocr9't^ 

%^o 

^^m 

) C^  70 

^. 

tC  I6J 

% 

CASE, 

2. 

To  reduce  numbers  of  different  denominations^  0ts  of  money  ^  weig/jt 
and  measure^  to  their  decimal  lvalues, 

RULE. 

"  1 .  Write  the  given  numbers  perpendicularly  urtdereach  Other  for  divi* 

"  dcnsls,  proceeding  orderly  IVom  the  least  to  the  greatest. 

L 


!2 


DECIMAL  FRACTIONS. 


Sect.  II.  2 


"  II.  Opposite  to  each  dividend,  on  the  left  hand,  place  such  a  number  for 
<'  a  divisor  as  will  bring  it  to  the  next  superior  denomination,  and  draw  a  line 
"perpendicularly  beiweenthem. 

"Ill,  Begin  with  the  highest,  and  write  the  quotient  of  each  division,  as 
"  decimal  parts,  on  the  right  hand  of  the  dividend  next  below  it,  and  so  on, 
"  tin  they  are  all  used,  and  the  last  quotient  will  be  the  decimal  sought." 

EXAMPLES. 

1.  Reduce  10*.  6|f/.  to  the  fraction  of  a  pound. 


OPERATION. 


3, 
6,75 


20  I  10,5625 


,528125  ^ns. 


The  given  numbers  arranged  for  the  op- 
eration, all  stand  as  integers.  I  then  sup- 
pose 2  cyphers  annexed  to  the  3  (3,00) 
which  divided  by  4,  the  quotient  is  75^ 
which  I  write  against  6  in  the  next  line  and 
the  sum  tiius  produced  (6,75)  I  divide  by 
12,  placing  the  quotient,  (5625)  at  the 
right  hand  of  the  10  :  lastly,  I  divide  by  20 
and  the  quotient,  (,528125)  is  the  decimal 
required. 


2.  Reduce 
mal  of  a  pound.  Am.  ,6r29-f- 


3^.  5|d.  to  the  deci- 


3.  Reduce   12pwts.  14grs.  to  the  de- 
cimal of  an  ounce.        Ans,  ,629 1. 


20 


\MsS3 
6  719+ 


2^ 


61?/+ 


CASE    5. 

To  find  the  value  ofanyghen  decimal  in  the  term^  of  an  integer  • 

RULE. 

IMuLTipLY  the  decimal  by  that  number,  which  it  takes  of  the  next  less  de- 
nomination to  make  one  of  that  deitoiunraion  in  which  the  decimal  is  given, 
and  cut  off  so  many  figures  for  a  remainder  to  the  ri,u,ht  hand  ©f  the  quotient, 
as  there  arc  places  in  the  given  decimal.  Proceed  in  the  same  manner  with 
the  remainder,  and  continue  to  do  so  thro*  all  the  parts  of  the  integer,  and 
the  several  denominations  standing  on  the  left  hand  make  the  answer. 


Sect.  II.  2.  DECIMAL  FRACTIONS.  83 

EXAMPLES. 

J.  WhatIs  the  value  of  ,528125  of  a 

pound  ?  This  question  is  the  first  example 

OPERATION.  in  the  preceding  case  inverted,  by 

,528125  which  it  will  be  seen,  that  questions 

2  0  in  these  two  cases  may  reciprocally 

prove  each  other. 

The  given  decimal  being  the  deci- 
mal of  a  peund,  and  shillings  being 
the  next  less  inferior  denomination, 
because20  shillings  make  one  pound, 
I  multiply  the  decimal  by  20,  and 
cutting  off  from  the  right  hand  of  the 
Farthings    5,0  0  0  0  0  0  product  a  number  of  figures,  for  a  re- 

^Tzs.  10s.   6|d.  mainder,  equal  to  the  number  of  fig- 

ures in  the  given  decimal,  leaves  10 
on  the  left  hand  which  are  shillings.  I  then  multiply  the  remainder  which  is 
the  decimal  of  a  shilling  by  12,  and  cutting  off  as  before,  gives  6  on  the  left 
hand  of  pence;  lastly,  I  multiply  this  last  remainder,  or  decimal  of  a  penny  by 
4  and  find  it  to  be  3  farthings,  without '  any  remainder.  It  then  appears  that 
,528125  of  a  pound  is  in  value  10s.  6|d. 

2.  What  is  the  value  of  ,73968  of        3.  What  is  the  value  of  ,7,68  of  a 
^ pound?        jins.jQ\4,  9lci.  -poundTroy  I  Jns. 9 ozAfiwt. 7 ^"grs. 

,75968  ■  .«! 


Shillings^  1 

©,5 

6 

2 

5 

0  0 

1  2 

PencCi 

6,7 

5 

0 

0 

0  0 
4 

/^.  79J60  ^'^^io 


; £  i%So 


il  18  the  iast  remainder,  ^80  reduced  to  it»  hweat  ferm».  J  fraction  is  said  t9 
be  reduced  to  its  lowest  terms,  vjhen  there  is  no  number  which  will  divide  both 
tht  numerator  and  denominator  without  a  remainder.  TiniSy  set  f  the  fraction 
Its  proficr  denominator  -f^o^,  then  divide  the  numerator  and  the  denominator  by 
any  number  which  will  divide  them  both  without  a  remainder,  continue  to  do  S9 
ts  long  as  any  number  can  be  found  that  will  divide  them  in  tfiaC  manner. 


PusS)^A\^^l-li 


S4/  SUrPLEMENT  TO  FRACTIONS.       Sect.  II.  2. 

Supplement  to  ^ptaCtlOltJ^t 

^-  ,M»      'f  "•  -JU-  ®  C-  v»  vU'^r  ■!■!    

QUESTIONS. 

1 .  JF/f  y<  y  are  fractions  ? 

2.  WiiAf  are  integers^  or  ivhole  numbers  ? 

3.  Wha'T  are  mixed  numbers  7 

4.  Cf  hcv)  many  kinds  are  fractions  ? 

5.  How  are  Vulgar  Fractions  written  ? 

6.  WHAris  sig7iified  by  the  denoTninat  or  of  a  fraction? 

7.  WnAr  is  signified  by  the  numerator  ? 

8.  //b«^  ore  Decimal  Fractions  written  ? 

9.  How  do  Decimals  differ  from  Vulgar  Fractions? 

10.  How  can  it  be  ascertained,  what  the  denominator  te  a  Decimal  Fraction  i^t 

if  it  be  not  exfir  eased? 

1 1 .  How  do  cyfihers  filaced  at  the  left  hand  of  a  Decimal  Fraction  affect  its  value  ? 

12.  How  are  Decimals  distinguished  from  whole  numbers  ? 

13.  /i^  the  addition  of  Decimals'  what  is  the  rulefsr  fiointing  off? 

14.  What  is  the  rule  for  fiointing  off  Decimals  in  Subtraction  ?  In  Multiplica- 

tion ?  and  in  Division  ? 
\S.  In  what  manner  is  the  reduction  of  a  vulgar  Fraction  to  a  decimal  fierf or  me  d? 

16.  How  are  numbers  of  different   denominations  as  Jiounds,  shillings,  pence, 

life,  reduced  to  their  decimal  values  ? 

17.  If  it  be  required  to  find  the  value  of  any  given  decimal  in  the  terms  of  an 

integer  what  is  the  method  of  procedure  ? 

EXERCISES, 

1.  What  is  the  sum  of  79-}  6iand  In  Case  1.  Ex.  3c?,  under  Re- 

of  J  when  added  together  ? 

Operation.  duction   of  decimal  fractions  the 

79,5 
6,25  Scholar  may  notice,  that  i,  I  and 

,75 
— ~-  I  reduced  to  decimals  are,  ,25  ,5 


86,50  Jins. 

and  ,75.    "When  numbers,  there- 
2.  FnoM  17  take  I 

operation.  fore,  for  operations  in    either  of 

,75  Ihe  fundamental  Rules,  are  in* 


16,25  Remainder.  cumbered  with  these  fractions  J^ 


Sect.  II.  2.       SUPPLEMENT  to  FRACTIONS. 


S5 


5.  Multiply  685^  by  5  | 

OPERATION. 

6  8,2  5 

5,5 


3  4  12  5 
1  2  5 


i.  Divide  26|-by  5^ 

OPXRATION. 

2,5)26,25(10,5   Quotient, 
25 


h  h  substitute  for  them  their 
equivalent  decimal  fractions, 
that  is,  for  a  ,25  for  g-  ,5  for 
1 ,75  then  proceed  according  to 
the  rules  already  given  for  these 
respective  operations  in  decimal 
fractions. 


Many  persons  are  perplexed  by  occurrences  of  a  Fimilar  nature  to  the  ex« 
fimples  above.  Hence  it  is  seen  in  some  measure  the  usefulness  of  Fractions, 
particularly  decimal  fractions.  The  only  thini^  necessary  to  render  any  per- 
son adroit  in  these  operations  is  to  have  riveted  in  his  mind  the  rules  for 
pointing  as  taught  and  explained  in  their  proper  places.  They  are  not  bur- 
thensome  j  every  scholar  should  have  them  perfectly  committed. 


5.  If   a  pile  of  wood   be  18  feet 
long,  ll|  wide,  and  7^  high,  how 
many  cords  does  it  contain  ? 
jins.  \2  cords  68  /eel*  432  inc/^cf^ 

IS 


9  0 

IS 

to  7,0 

f03SO 

l^^9  0 


6^ 


A  CORD  of  wood  is  128  solid 
feet;  the  proporlionscommonly  as- 
signed arc,  8  feet  in  length,  4  in 
breadth,  and  4  in  height. 

The  contents  of  a  load  or  pile  of 
wood  of  any  din.ensions  may  be 
found  by  multiplying  the  length  by 
the  breadth  and  this  product  by  the 
height  ;  ow,  by  multiplying  the 
length,  breadth,  and  height  into 
each  other.  The  last  product  di- 
vided by  128  will  shew  the  number 
of  cords,  the  remainder,  if  any,  will 
J)e  so  many  solid  feet. 

XO  00 
^0  0 


J3Y 


trt^/tJ*i. 


The  432  inc/ica  in  thcfvactiony  ,25  of  a  foot  valued  accordirff  to  Case  3,  lie 
due.  Dec.  fractions. 


36 


SUPPLEMENT  TO  FRACTIONS.      Sect.  II. 


6.  If  a  load  of  wood  be  9  feet  long         7.  What   is    the  value   ,725 
S^  feet  wide,  and  4  feet  high,  how  nia-     day  ? 
ny  square  feet  does  it  contain  )  ^ns.  17  hours  24  minutes, 

f4ns.  126 feet,  which  is  2  feet  short  of 
a  cord. 


i± 


4'y 
1^ 


5  IS 

JL    , 

I  16jO  :J/^. 


17,4^0  0 
dO 


V,000 


«.  What  is  the  value  of  ,0625  of 
%  shilling  ?         Ans,  3  farthings. 


9.  RjiDucE  ^Cwt.  Qqrs.  7lh.  ^oz.  ta 
the  decimal  of  a  Ton. 
4ns.  ,l5334821-{. 

lb  8, 

iH).2bn^7f\ 


10. 


10.  Reduce  3  farthing;sto  the  deci-        1 1.  Reduce  ^^  to  a  decimal  fracn 
Kial  of  a  shilling.  *^ws.  ,0625  tion.  Jns.  jQ\25. 


/2L 


3. 


ISLOO 
2.^00 


Sect.  II.  3.  FEDERAL  MONEY,  87 

§  3.  ftnttai  Mmtp, 

-^mmm  'Ai  -:::-  -:'.5-  >|c  ^>  •»/.;-  -i:?-  <^ — 

Federal  Money  is  the  coin  of  the  United  States,  established  by  Congrcss> 
A.  D.  1786.  Of  all  coins  this  is  the  most  simple,  and  the  operalions  in  it,  the 
most  easy. 

The  denominations  are  in  a  decimal firofiortkn^  as  exhibited  in  the  following 

TABLE. 

10  Mills      )  C  Cent, 

10  Cents     C        1.  )  Dime, 

10  Dimes  C  "laKeone  \  DoJ,^,.^  marked  thus,  % 

10  Dollars  )  C  Eagle, 

The  expression  of  any  sum  in  Federal  Money  is  simply  the  expression  of 
a  mixfd  number  in  decimal  fractions.  A  dollar  is  the  Unit  money  ;  dollars 
therefore  must  occupy  the  place  of  units,  the  less  denominations,  as  dimes, 
cents,  and  mills,  are  decimal  parts  of  a  dollar,  and  may  be  distinguished 
from  dollars  in  the  same  way  as  any  other  decimals  by  a  comma  or  separatrix. 
A'l4  tlie  figures  to  the  left  hand  of  dollars,  or  beyond  Units  place  arc  eagles. 
Thus,  17  Eagles,  5  dollars,  3  dimes,  4  cents,  and  6  mills  are  written — 

V.     V. 

^  ^  -^  <;         Or  these,  four  are  real  coins,  and  one  is  imagin- 


a  -c  -^i 


S   R 


"Q  "2      ary. 


^1     K 


C  '^  -:  ^  -^  §  ^"^'  ^^^^  ^^^"^  ^^®  ^^^  Eagle,  a  gold  coin  ;  the 

^  ^  '^  §  o      Dollar  and  the  Dime,  silver  coins  ;  and  the  Cent,  a 


tSs^^ 


li^  g  copper  com.    The  mill  is  only  imaginary,  there  bc- 

"<    o  o  C  ;:7  i"g  no  piece  of  money  of  that  denomination 
,"     •-  ••>  f  °  There  are  half-eagles,  l;alf-dollars,  double-dimes, 

^  ."°  S  ^^  ^  half-dimes,  and  half  cents,  real  coins. 


'^       o    ^    -     - 

17    5,  3  4   6 


These  denominations,  or  different  pieces  of  money,  being  in  a  tenfold 
J»roportion,  consequently,  any  sum  in  Federal  Money  does  of  itself  exhibit 
tiie  particular  number  of  cacli  diffcrci»t  piece  of  money  contained  in  it.  Thus' 
175,346  (neventfen  cuglea.Jive  dollars,  three  dimes,  four  cents,  six  miUs)  con- 
tain 175346  mills,  17534 -^6,  cents,  1 75 3^^^^  dimes,  1 7 5-,^/^,  dollars,  17-,%3,*«, 
eagles.  1  herelore,  eagles  and  dollars  reckoned  together,  express  the  num- 
ber of  dollars  contained  in  the  sum  ;  tke  same  of  dimes  and  cents  ;  and  thia 
indeed  IS  the  usual  way  of  account,  to  reckon  tfee  whoio  sum  in  dollars,  cents, 
and  mills  thus,  ' 


S  175     34    G 


_  Wl  Addition,  Subtraction,  Mwltiplication  and  Division  of  (Federal  Money 
Us  performed  in  all  respects  as  in  Decimal  Fraciions,  to  which  the  Scholar  it 
f referred  lor  the  use  of  rules  in  these  operations. 


83         ADDITION  or  FEDERAL  MONEY.      Sect.  II.  3. 


ADDITION  OF  FEDERAL  MONEY. 

1.  Add  16  Eagles  ;  3  Eap^les,  7  Dollars,  5  Cents;  26  Dollars,  6  Dimes,  4 
G;iius,  3  Mills  ;  75  Cents,  8  Mills  j  40  Dollars,  9  Cents  together. 
Operation. 

Or,  the  sums  may  be  reckoned  in  dol- 
lars, ce;its,  and  mills,  thus, 


Dime 

Cts. 

1 

/^-^^ 

16  0, 

3   7, 

0  5 

2  6, 

6  4 

3 

* 

7  5 

8 

4  0, 

0  9 

^26  4, 

5  4 

1 

1  a 

^ 

^ 

160 

37  05 

26  64 

3 

75 

8 

40  C9 

S264  54     1 

2.  If  I  am  indebted  59  dollars,  1 12  dollars,98  cts.  113  dolls.  15  cts,  15  dolls. 
21  dolls.  50  cts.  200  dolls.  73dolls.  35  dolls.  17  cts.  75  dolls.  20  dolls.  40  dolls, 
33  cts.  and  IG  dolls.     What  is  the  sum  v/hich  I  owe  ? 

jins.   1^781      13 

/  /  ^  9^S^  AticouNTANTs  generally  omit  the 

/  /  3  /  "T"  comma  and  distinguish   cents    from 

,  j\.  dollars  by   setting  them  apart  iVora 

L'l  iW}  the  dollars. 

ZOO 
J6 


T78I     t2>  u^i^. 


Segt.  II.  3.   SUBTRACTION  of  FEDERAL  MONEY.   89 


SUBTRACTIOIJ  OF  FEDERAL  MONEY. 


1.  From  Dolls.  863,17 
take,  Dolls.  69,82 

OPERATION. 

8  6  3,1   7 
6  9,8  2 


Remainder  J  7  9  3,3  5 


2.  From  Dolls.  681 
take,  Dolls.  57,63 
Remainder ^  DolL  62  3,37 

6Sh  00 


MULTIPLTCATION  OF  FEDERAL  MONEY. 
1.  If  Flour  be  Dolls.  10,25 //<fr  barrel,  what  will  27  barrels  costs  ? 


OPEKATION. 

1   0,2  5 
2   7 

7   17  5^ 

2  0  5  0 


Doll's.  2  7  6,  7  5  Answer, 
2.  Multiply  Dolls.  76,35 
by  Dolls.  37,46 
Product,  Brolb.  2860,0710 


I Q  6,  op  no  o^. 


Point  off  the  decimals  in  the  pro- 
duct according  to  the  rule  in  multi- 
plication of  decimals  ;  if  at  any  time 
there  shall  be  niore  than  three  deci- 
mal figures,  all  beyond  mills,  or  the 
third  place,  will  be  decimal  parts  of 
a  ttiiii. 

3.  Multiply  Dolls.  24,675  by 
Dolls.  13,63. 
Product.  Dolls,  336,320  -^^^ 


/4S  0  ^O 

TocT 


DIVISION  OF  FEDERAL  MONEY, 
I.  It  ^!r28  bushels  of  wheat  cost  i?o//*.  3961,  how  much  is  it  per  bushel  ? 


OPERATION. 

Bushpls.  Dolls.  D.  d.  c.  m. 

2728)2961(1,  0  8  5  Amwer. 
2728 

23300 
21824 


U760 
13640 


When  the  dividend  consist  of  dol- 
lars only,  if  there  be  a  remainder  af- 
ter division,  cyphers  must  be  annex- 
as  in  division  of  decimals. 


1120 


M 


90 

2.  Div] 
13  men  ; 


DIVISION  OF  FEDERAL  MONEY.     Sect.  II.  3, 


DE  Dolls.  375  6  equally  among 
what  will  each  man  receive  ? 
^ns.  Dolls.  288,923. 

26 

~77T 
104 
lie* 

10^ 


27. 


3.  Divide    Dolh.     16,75   by 
Product^  62  cents. 


IZC 

iir 

30 
26 


39 


REDUCTION  OF  FEDERAL  MONEY'. 
CASE     1. 

To  reduce  Pounds,  Shillings,  Pence  and  Farthings,  to  Dollars^ 
Cents  and  Mills-, 

RULE. 
Set  dovni  the  pounds  and  to  the  right  hand  write  half  the  greatest 
evennjmber  tlie  given  shillings  ?  then  consider  how  many  farthings  tlitre 
are  contained  in  the  given  pence  and  farthings,  and  if  the  sum  exceed  12, 
increase  it  by  1,  or  if  it  exceed  36  increase  it  by  2,  which  sum  set  down  to  the 
ri;2,ht  hand  of  half  the  greatest  even  number  of  shillings  before  written,  re- 
membering to  increase  the  second  place,  or  the  place  next  to  shillings  by  5, 
if  the  sliillingsbe  an  odd  number  ;  to  ihc  whole  sum  thus  produced,  annex  a 
cypher  and  divide  the  sum  by  3  ;  cut  off  the  three  right  haiid  figures  in  the 
quotient,  which  will  be  cents  and  mills,  the  rest  wfll  be  dollars. 

EXAMPLES.  t 

1.  Reduce  ^' 47  7^.  I Ojc/.  to  Dollars,  Cents,  and  Mills. 

OPERATION. 

In   this    example   to   the  right   hand   of 
pounds  (47)  I  write  3,  half  the  greatest  even 
,   C  number  of  the  given  shillings  (7,)  the  far- 

things in  \0^d.  (43)incseaseii  by  two(45)be- 
cause  exceeding  36  and  the  second  place  in- 
creased by  5  because  shillings  were  an 
odd  number,  are  95,  which  sum  written  to 
tlie  right  hand  of  the  3,  a  cypher  annexed, 
and  the  sum  divided  by  S^gives  the  ^lns-ii>cr.^ 
157  dollars,  98  cents,  and  o^niltn. 


l§s 

"■^   s  '^ 

•^    <3    fco 

•s  ^.s 

^   "v3 

O^  ^    o 

i^  -^^ 

^  ^  «^ 

s  ~  •?  ^ 

a  ?  ^  H 

e< 

.    «    t^  C3     ii 

"5  :S  S  S 

O 

V  -  •-    ;^ 

fis 

^^  <  ^   ^ 

<4j 

»§  '■==  -J  >. 
^  ?^  ^  v^ 

/^A.^^      r'.-\>-> 

DrviDE  by  3)  4  : 

r  3  9    5  0 

Do! 


1    5   7, 9   8 


Sect.  II.  3.     REDUCTION  of  FEDERAL  MONEY.       91 


Jf  fiounds  only  are  given  to  be  reduced^  a  cypher  must  be  annexed  and  the 
number  divided  by  3  :  the  quotient  will  be  dollars.  If  there  be  a  remainder 
annex  more  cyphers,  and  divide,  the  quotient  will  be  cents  and  mills. 

TVhen  there  are  no  shillings^  or  only  1  shilling  in  the  given  mim^  so  there  be  no 
even  number^  w\\\G -A  cypher  \x\  place  or  half  the  even  nurpiber  of  shillings, 
then -proc&ed  with  the  pence  and  farthings  as  in  other  cases* 

If  it  he  required  to  reduce  fiounds^  shillings^  fience^  iD'c.  to  Dollars  and  cents 
only,  the  cypher  must  not  be  annexed  ;  in  this  case  two  figures  only  must 
be  cut  oif  from  the  quotient. 

A  LITTLE  practice  will  make  these  operations  extremely  easy. 


2.   lN;C"r63  how  many  dojiars, 
cents  and  mills  ? 

Am.  2543  Dolls.  33  cts,  3  m. 

2))r62)OO00 


3.  In  £\7   \s.  &\d.  hew  many  dol- 
lars, cents  and  mills  ? 

Ans.  Dolls.   5-6  92  3. 

3)  irorr 


4-  In  ^109  S*.  8cf.  how  many 
(dollars,  and  cents  ? 

Jns.  Dolls.  363,944.4- 

^6  5,9  ^7? -i- 


5.  In  ;C86  6*.  5^r/  how  many  dol- 
lars, c€»its  and  mills  ? 

Ans.  Dolls.  287,74. 

2<9  z  r4~ 


CASE    2. 

To  reduce  Dollars,  Cents ^  and  Mills,  to  Pounds,  SbilUngs,  Pence, 

and  Farthings, 
RULE. 

..'^rv^^  !^'^  ?'"'" '"""  '^>^  ^' "'"'  °"  ^'^^  ^«"^  '-'^'^  ''--^"^^  ^'^urcs,  which 

will  be  decimals  of  a  pound,  the  left  hand  figures  will  be  the  pounds.  To  f.nd 
the  value  of  the  deiimals,  douhle  the  first  figure  for  shillings,  «nd  if  the  figure- 
in  «^^y;^"^,P  "ce  be  5,  add  another  shilling,  then  call  ihe  figures  in.thc  sec  - 
ond  and  hird  places,  af^er  deducting  the  5  in  the  second  place,  so  many  far- 
things,  abaiu^  1  when  they  arc  above  12,  and  2,  when  Ihcy  arc  above  5G. 


92    REDUCTION  OF  FEDERAL  MONEY.     Se<:t.  If.  3. 


JEXAMPLIIS. 


1.  Reduce  255  dollars  40  cents,  6  mills,  to  pounds,  shillings  pence  andj 
farthings. 


OPERATIOK. 
255406 


•■•'      — -^ 
76]6216 
r^C/'e    12.S.  5(f.  Jinsiver. 


In  this  example  having  mnlti plied  t1i€ 
given  sum  by  3  and  cut  off  the  four  right 
band  figures  of  the  product,  I  double  the 
first  figure  (6)  for  shillings,  the  figures  in 
the  second  and  third  places  (21)  abating 
1  for   being   over    12,(20)  I  consider  as 
farthings,  equal  to  5c?.     In  Dolts.  255,40^ 
therefore,  are  ;C'^6   Igs  5d. 
Here  8  in  the  fourth  place  of  decimals  (tt^Ioo  of  a  pound)  beinj^pf  infe-* 
rior  value  is  not  reckoned.     The  loss  in  this  place  is  always  less  than  one  tar- 
thing. 

If  there  be  no  mills  in  tike  given  «Mm,  multiply  as  before  and  cut  off  Sj^gure* 
only. 

If  there  be  neither  cents  nor  milU  that  is,  if  the  given  sum  be  cfo//flr«,  multiply 
by  3  and  cut  off  one  figure  only. 

2.  In  Dolls.  392,75  how  many  pounds         3.  In  Dolls.   39,635   how  msiny 
shillings,  pence  and  farthings  ?  pounds,  shillings  pence,  Uc.  ? 

4n8,  ^117  'l6«.  6rf.  4n8.  f;  \\    17«  9\d, 


£u.t7.9i 


4.  Reduce  134  dollars,  65  cenisto 
pounds,  shillings,  pence,  and  farthings. 

Ms,  £40  7s.   \Old, 

WSJT 


5.  Redug*  Dolls.  684  to  pounds 
and  shillings,    Ans.  >^205  4«» 


G8^ 
3 


Sect.  II.  J.    SUPPLEMENT  to  FEDERAL  MONEY.     93 

Supplement  to  5f0t!trai  ^Ofitp. 

QUESTIONS. 

).  WhaTis  F^DRRAJL  MoNEr  ^   When  was  its  establishment,  and  by  v^hat 

authcriiy  ? 

3.  IVfiAtare  the  denominations  in  Federal  Money  ? 

3.  Which  is  the  Unit  Money? 

4.  How  are  dollars  distinguiahedfrom  dimes,  cents  and  jniils  ? 

5.  Wha  rfilaces  do  the. different  denominations  occufiy,from  the  decimal t^oints^ 

6.  Ho IV  is  the  Addition  of  Federal  Money  pei'formed?    Subtraction?  Mulf 

tifilication  ?  Division  ? 

7.  By  what  method  are  Pmmds,    Shillings,  Pence  and  Farthings  reduced  to 

Federal  Money  7 
S.  Hqw  ctre  Dollars^  JDimes,   Cents  Qnd  Mills,   reduced  to  Founds^  Shillings 
Fence  and  Farthings  ? 

FXERCISES. 

1.  A  WAN   dies  leaving  an  estate  of  2.  A  man  sells  1225  bushelfcof 

71600  Dollars,  there  are  demands  a-  "wheat  at  Dol.  1,33  per  busl-wel,  and 

gainst  the  estate  of  Dol.  39876,74  ;  the  receives  Dol. 93, 76  for  transporta- 

residue  is  to  be  divided  between  7  Bons  J  tion  ;  whatdoeshe  receiveiulhc 

^vhat  mil  each  one  receive  ?  -whole  ? 

Jn9.  4531  Volls.  89  c(s,  4n8,  Dolls,  1723,01. 

i9  8r6.r//  ^,  A 5 5   ^ 

TH I  715.16  ^{^ 

^^3)  .89.5  .4^.      /fi^^ 


T 71^.0)  'M. 


94     SUPPLEMENT  to  FEDERAL  MONEY.  Sect.  IL  3, 


3.  Reduce  yC  "75  Is.  e^rf.  to  Dol-         4.  In /7  13*.  8cL  how  many  dollars 
I^is  and  cents,  Jus.  Dolls.  1250,2  56     cents  and  juills  ?    j^ns.  Bolls.  25  61 


3}161A 


ZC,6/ 


5.  Reduce  Dolls.  781,27  to  pounds       6.  Reduce  Dolls.  98,763  to  pounds 
phillings,  pence,  and  farthings*  shillings,  pence,  and  farthings. 

Ms.  £23i  7s.  r^d,  Jns.£  29  12s.  7d. 


TABLE 

For  reducing  Shillings  and  Pence  to  Cents  and  Mills, 

Shill.  Shiil.  Shill.  Shill.  Shili. 

12  3  4  5 


0 


Pence 
0 

1 

Cts.Mills 
I          4 

Cts.Mills 
16       7 

18        1 

Cts.Mills 

33  3 

34  7 

Cts.Mills 
50 

51        4 

Cts.Mills  1 
66        7 

68        1 

Cts.MilU 

83  -3 

84  7 

2 

2 

8 

19       5 

36 

1 

52 

8 

69 

5 

86        1 

3 

4 

2 

20       9 

37 

5 

54 

2 

70 

9 

87       5 

4 

5 

6 

22       3 

38 

9 

55 

6 

72 

3 

88       9 

5 

7 

23        7 

^40 

3 

57 

73 

7 

90       3 

6 

8 

3 

25 

41 

1 

58 

3 

75 

91       6 

4 

9 

7 

26.## 

43 

59 

7 

76 

4 

93 

8 

11 

k 

^7    'Hi 

►44 

4 

61 

1 

77 

8 

94       4 

9 

10  . 

12 

13 

5 
9 

2^9-^2 
.  30  .     6 

45^ 

47 

8 
2 

6^ 
63 

5 
9 

79 
80 

2 
6 

95  8 
97       2 

11 

15 

3 

32    "' 

48 

6 

65 

5 

82 

igS       6 

To  find  by  tliis  Table  the  Cents  and  Mills  in  any  sum  of  Shillings  and 
Pence  under  one  dolhr,  look  the  Shillings  at  top,  and  the  Pence  in  the  left 
hand  column,  then  under  the  foriner  and  on  a  line  wilh  the  latter,  will  be 
found  the  Cents  and  Mills  sougist* 


Sect.  II.  ^.  Table  for  reducing poundsy  ^c,  to  dollars y  ^c»       95 

TABLE. 

For  rsducing  the  Currencies  of  the  sender al  United  States   to    Fed* 

eral  Money, 


N.  Hamp. 

N.  Jersey. 

Mass. 

N.  York. 

Pennsylvania 

S.  Carolina. 

Rh.  Island, 

and 

Delaware, 

and 

Conn,  and 

N.  Carolina. 

and 

Georgia. 

Virginia. 

Maryland. 

D.cis.m. 

D.cts.iii. 

D.cts.in. 

1  D.cu.m. 

:f       1 

,       3 

,         3 

,        3 

,        4 

1       2 

,       7 

,        5 

,        6 

,         9 

,      3 

,    IQ 

,        8 

,        8 

,     14 

1 

,    14 

,     10 

,      il 

,     18 

2 

,    28 

,     21 

,     22 

,       36 

3 

,    42 

,     31 

,     33 

,     54 

4 

,    5  6 

,     42 

,    44 

,     7  1 

5 

,    6  9 

,     52 

,     56 

,     89 

.•   5 

,    83 

,     62 

,     ^7 

,107 

,    97 

,     73 

,     78 

,125 

8 

,111 

,     8  3 

,     89 

,143 

9 

,12  5 

,     94 

,100 

,161 

10 

,13  9 

,104 

,111 

,179 

II 

,153 

,114 

,122 

,196 

=      1 

,167 

,125 

,133 

,214 

2 

,3  3  3 

,250 

,267 

,429 

3 

,5  0  0 

,375 

,400 

,643 

4 

,6  6  6 

,500 

,533 

,857 

5 

,833 

,625 

,66  7 

1,171 

6 

1,0  0  0 

,750 

,800 

1,286 

7 

1,167 

,875 

,9  33 

1,500 

'          8 

1,3  3  3 

1,000 

1,067 

1,714 

9 

1,500 

1,125 

1,200 

J, 929 

10 

1,6  6  7 

1,250 

1,333 

2,143 

H 

1,833 

1,37  5 

1,467 

2,6  5  7 

12 

2,0  0  0 

l,5oO 

1,600 

2,5  71 

13 

2,16  7 

1,62  5 

1,  733 

2,786 

14 

2,3  3  3 

1,750 

1.8(57 

3,oog 

■ 

15 

2,500 

1,875 

2,000 

3,214 

16 

2,6  6  7 

2,  0  0  0 

2,  l33 

3,424 

17 

2,8  3  3 

2,  125 

2,267 

o^e'i^ 

18 

3,000 

2.  2  5  0 

2,400 

3,857 

19         / 

3,16  7 

2,3  75 

2.53  3 

4,071 

96     Table  far  reducing  Pounds ^  ^&.  to  Dolhirs,  Wc.  Sect.  II. 

TABLE 

For  reduc'wg  the  Currencies^  ^c.  continued. 


1 

2 

4 
S 

6 

7 

8 

9 

10 

20 
30 
40 
50 
60 

ro 

80 
90 
100 
200 
3D0 
400 
500 
6D0 
7-00 
8100 
900 
1.000 


New-Hamp. 

New-York, 

New- Jersey, 

South-Caroli. 

S:c.&c. 

&c. 

8CC.&C. 

8cc. 

D.c.m. 

D.c.m. 

D.c.m. 

D.c.m. 

:'     3,333 

2.5 

2,666 

4,286 

1      6,667 

5,0 

5,333 

8,571 

10,000 

7^5^ 

8,000 

12,857 

'     13,333 

10,0 

10,667 

17,143 

10,667 

12,5 

13,333 

21,429 

20,000 

15,0 

16^000 

25^714 

33,333 

17,5 

18,667 

30,000 

26,667 

2(^,0 

21.,333 

34,286 

30,000 

22,5 

24,000 

38,571 

3^,^33  ' 

2«^,0 

26,^67 

42,8  57 

6&,&67 

50,0 

53.333 

85,714 

100,000 

75,0 

80,000 

128,571 

133,333 

100,0 

106,667 

171,429 

166,667 

125,0 

133,333 

214,286 

200,000 

150, 

1  60,000 

257,143 

233,333 

175, 

186)667 

300,000 

256,667  , 

200, 

2131333 

342,85  7 

300,000  ' 

225, 

240,000 

385,714 

333,333 

250, 

266,667 

428,571 

666,667 

500, 

533,333 

857,143 

lOOO.OOO 

750, 

800,000^ 

1285,714 

1333,333 

1000, 

1056,667 

1714,286 

1666,567 

1250, 

1333,333 

2142,857 

2000,000 

1500, 

1600,000 

25  7r,429 

3333,335 

1750, 

1865,667 

3000^000 

2656, 6£6 

2000, 

2133,333 

3428,571 

3000,000 

2250, 

24.00,000 

385  7,143 

3333,333 

25.00,   1 

2666,667 

4285,714 

^or  reducing  Federal 


I  New-Hamp.  | 
I  £cc.  ^c>  I 
I       Dol.6/;       I 


TABLE 
Money  to  the  currencies  of  the  several 
Ujiiied  States, 
New- York,    j  Nevy-Jersey,  ['  South-CaroU. 
&c.  I      8cc.  &c.  Sec. 

Dol.  Bf.         I     Pol.  7f6.  Dol.  4/8. 


D.  Ct3.    1 

£.  s.d.q;   1 

£.  s.d.q^ 

!  ^.s.d.q. 

^.  s.d;q. 

,01 

o 

1.  0 

2 

,02 

1  2 

2  0 

1  0 

,03^ 

2  1 

3  0 

2  3 

1  3 

,04- 

3.   0 

S  3 

S  2 

3  1 

,05 

3  2 

4  3 

4r  2 

2  3 

,06 

:      *  1 

5  3 

5.  2 

3  1 

,0.7- 

5,0 

C  3 

6  1 

4  0 

vOS 

5  3 

7  3 

7  1 

4  2 

,0.9' 

6  2 

8:  3 

8  0 

5-0 

,10 

7  1 

9  2 

9  0 

5  2 

Sect.  II.  5.    Table  for  reducing  Dollars,  ^c,  to  Pounds,  ^c.  97 


TABLE 

For  reducing  the  currencies,  ^c.  continued. 


New 

■Hamp 

New 

-York, 

New- 

Jersey, 

South 

Cai 

olinai 

8cc 

.  &c. 

&c. 

Sec. 

&c. 

£cc 

3. 

d.  q. 

Dolls,  cts 

£ 

s.d.q. 

£- 

s.  d.  q. 

£ 

.  s.  d.  q. 

£' 

,20 

1  2  2 

1  7  1 

1  60 

11  1 

.30 

1  9  2 

2  4  3 

2  3  0 

I 

4  S 

,40 

2  4  3 

3  2  2 

3  0  0 

1 

10  2 

,50 

3  0  0 

4  0  0 

3  9  0 

2 

4  0 

,60 

3  7  1 

4  9  2 

4  6  0 

2 

9  2 

,70 

4  2  2 

5  7  1 

5  3  0 

3 

3  1 

,80 

4  9  2 

6  4  3 

'6  0  0 

3 

8  3 

,50 

5  4  3 

7  2  2 

6  9  0 

4 

2  2 

1, 

6  0  0 

8  0  0 

7  6  0 

4 

8  0 

2, 

12  0  0 

16  0  0 

15  0  0 

9 

4  0 

3, 

18  0  0 

1 

4  0  0 

1 

2  6  0 

14 

0  0 

4, 

1 

4  0  0 

1 

12  0  0 

1 

10  0  0 

18 

8  0 

5, 

1 

10  0  0 

2 

0  0  0 

1 

17  6  0 

1 

3 

4  0 

6, 

1 

16  0  0 

2 

8  0  0 

2 

5  0  0 

1 

8 

0  0 

7, 

2 

2  0  0 

2 

16  0  0 

2 

12  6  0 

1 

12 

8  0 

8, 

2 

8  0  0 

3 

4  0  0 

3 

0  0  0 

1 

17 

4  0 

9, 

2 

14  0  0 

3 

12  0  0 

3 

7  6 

2 

2 

0  0 

10, 

3 

0  0  0 

4 

0  d  0 

^ 

15  0 

2 

6 

8  0 

20 

6 

8 

7 

10  0 

4 

13 

4 

oO 

9 

12 

11 

5  0 

7 

0 

0 

40 

12 

16 

15 

0  0 

9 

6 

8 

50 

15 

20 

18 

15  0 

11 

13 

4 

60 

18 

24 

22 

10  0 

14 

0 

0 

70 

21 

28 

26 

5  0 

16 

6 

S 

80 

24 

32 

30 

0  0 

18 

13 

4 

90 

27 

36 

33 

15  0 

21 

0 

0 

100 

30 

40 

37 

10  0 

23 

6 

8 

200 

60 

80 

75 

0  0 

46 

13 

4 

300 

90 

120 

112 

10  0 

70 

6 

0 

400 

120 

160 

150 

0  0 

93 

6 

8 

500 

150 

200 

187 

10  0 

116 

13 

4 

600 

180 

240 

225 

0  0 

140 

0 

0 

700 

210 

280 

262 

10  0 

1C3 

6 

8 

800 

240 

320 

300 

0  0 

186 

13 

4 

900 

270 

360 

337 

10  0 

210 

0 

0 

1000 

300 

400 

575 

0  0 

333 

6 

8 

2000 

600 

800 

750 

466 

13 

4 

r,ooo 

900 

1200 

1  125 

700 

0 

0 

4000 

1200 

1600 

1500 

9  33 

6 

8 

5000 

1500 

2000 

1875 

1  166 

1  :1 

A. 

6000 

18(A) 

2i00 

2250 

Uo.» 

7000 

2  100 

2800 

2625 

165:1 

8(K)0 

2. 100 

3200 

5000 

l8t*M"i 

yooo 

2700 

5600 

.«  .  —  ,. 

'  1  (  .(  : 

10000 

3000 

4000 

N 


98  SIMPLE  INTEREST.  Sect.  II.  4. 

§  4..  mmm. 

Interest  is  the  allowance  given  {or  the  use  of  money,  by  the  Borrower  to 
the  Lender.  It  is  computed  at  so  many  dollars  for  each  hundred  lent  for  a 
year,  ffiey-  annum)  and  alike  proportion  for  a  greater  or  less  time.  The  high- 
est rate  is  limited  by  our  laws  to  6  per.  cent,  that  is  6  dollars  for  a  hundred 
dollars,  6  cents  for  a  hundred  cents,  ^6  lor  a  ;ClOO,  Sec.  This  is  called  le^aL 
interest^  and  is  always  understood  when  no  other  rate  is  mentioned. 

There  are  three  things  to  be  noticed  in  Interest. 

1.  The  Principal  ;  or,  money  lent. 

2.  The  Rate  ;  or,  sum/ztr  cent^  agreed  on. 

3.  The  Amount  ;  or,  Principal  and  Interest  added  together. 
Interest  is  of  two  sorts,  simfile  and  comfiound. 

1.  Simple  Interest  is  that  which  is  allowed  for  the  principle  only. 

2.  Compound  Interest  is  that  which  arises  from  the  interest  being  added 
to  the  principal  and  (coniinuing  in  the  hands  of  the  lender)  becomes  apart 
of  the  principal,  at  the  end  of  each  stated  time  of  payment. 

GENERAL  RULE. 

1.  For  one  year.,  multiply  the  principal  for  the  rate,  from  the  product  cut 
off  the  two  rii^ht  hand  figures  of  the  dollars,  which  will  be  cents,  those  to  the 
left  hand  will  be  dollars  ;  or,  which  is  the  same  thing,  remove  the  nefiaratrix^ 
from  its  natural  place  two  fij^ures  towards  the  left  hand,  then  all  those  fig- 
ures to  the  left  hand  will  be  dollars,  and  those  to  the  light  hand  will  be  cents, 
luills,  and  parts  of  a  mill. 

In  the  same  way  is  calculated  the  interest  on  any  sum  of  money  in  fioxinds.^  shil- 
lings., pence  and  farthings^  ivith  this  difference  only.,  that  the  two  figures  cut  off 
to  the  right  hand  of  pounds.,  must  be  reduced  to  the  Ictuest  derio-ndnatiovy  each 
time  cutting  off  as  at  first. 

2.  For  t%vo  or  more  years,  multiply  the  interest  of  one  year  by  the  number 
of  years. 

3.  For  months,  take  proportional  or  aliquot  parts  of  the  interest  for  1  year 
that  is,  for   6  months,  -^-  ;  for  4  months,  |  ;  for  3  months,  i,  Sec. 

4.  For  days,  the  proportional  or  aliquot  parts  of  tiie  interest  for  1  month, 
allowing  30  days  to  a  month. 

EXAMPLES. 
1.  What  is  the  interest  o^ Bolls.  86,446  for  one  year,  at  6  per  cent  ? 

OPERATION. 

JDolls.  cts.  m.  Ix  the  product  of  the  principal  mulli- 

86,  44  6  principal.  plied  by  the  rate  is  found  the  answer. 

6  rate.  Thus,  cutting  off  the  two  right  hand 

figures  from  the  dollars  leaves  5  on  the 

5/18,67  6  Interest.  left  hand  which  is  dollars  ;  the  two  fig - 

•    _        _  ures  cut  off  (18)  are  cents,  the  next  Cg- 

nre  (6)  is  mills,  all  the  figures  which  may  chance  to  be  at  the  right  hand  of 

mills,  are  parts  of  a  iiiill,  hence  we  collcct'the  Jno-vcr.  5  /Jo//?.  \^'cts.  6  -{^m. 


Sect.  II.  4.  SIMPLE  INTEREST.  99 

2.  What  is  tlte  intefeSt  of  Dolls.  365   14c/«.    6m///5,  for  three  years,  7 
months  and  6  duys  ? 

OPERATION. 

3  6  5,    1   4   6  firincifial. 
6  rate. 


6  Months  ^)2   1[  9  0,  8  7   6  interest  for  one  year. 
3 


6  5,   7  2  6  2  8  interest /•r  Z  years. 
1   Month  |)1  0,  9   5   4  3   8  interest  for  6  months. 
6  Days  -5)  I,  8  2  5  7  3  interest  for  I  month. 
,36514  interest  J  or  6  days. 


7  8,  8  7  15  3  interest  for  3  years,  7  months,  and  6  days  ; 
that  is  78  dolls.  BTcts.    l-f-^\m. 

Because  7  months  is  not  an  even  part  of  a  year,  take  two  such  numbers 
as  are  even  parts  and  which  added  together  will  make  7  (6  and  1,)  6  months 
is  -|  of  a  year,  therefore,  for  6  months,  divide  the  interest  of  one  year  by  2  ; 
«gain,  1  month  is  |  of  6  months,  therefore  for  1  month^  divide  the  interest  of 
6  months  by  6.  For  the  days,  because  6  days  is  |  of  a  month,  or  of  30  days, 
lhcrefore,ycr  6  days,  divide  the  interest  of  1  month  by  5.  Lastly  add  the  in- 
terest of  all  the  parts  ©f  the  time  together,  the  swm  is  the  answer. 

3.  What  is  the  interest  of  ^71  7s.  6^d.  4.  What  is  the  interest  of  16s. 
for 


year,  at  6  per  cent 

OPERATION. 

£.     s.     d.     q. 
71      7     6     2 
6 

fid.  for  1 

year  ? 

Ans. 

s 

6 

S.0 

0 

0 

^4  28      5 

20 

3 

0 

^/,oo 

«.  5165 
12 

d.  7(83 

4 

q.  3)32  Ans.^  4   5*.  1\d. 


100^  SIMPLE  INTEREST.  Sect.  II.  4. 

When  the  rate  is  6  per  cent,  there  is  notj  perhafis,  a  more  concise  and  easy 
way  of'casting  interest^  on  any  su7n  of  money  in  Dollars^  CentSy  and  Millsj  than 
by  ihe  following' 

METHOD, 

Write  down  half  the  greatest  even  number  of  months  for  a  multiplier  ;  if 
there  be  an  odd  month  it  mast  be  reckoned  20  days,  for  which  and  the  given 
days,  if  any,  seek  how  many  times  you  can  have  6  in  the  sum  of  them,  place 
the  figure  for  a  decimal  at  the  right  hand  of  half  the  even  number  of  months, 
already  found,  by  which  multiply  the  principal  ;  observint^  in  pointing  off  the 
product,  10  remove  the  decimal  point  or  separatrix  tn^o  figures  from  its  natur- 
al place  towards  the  left  hand,  that  is  point  off  t'lvo  more  places  for  decimals  in 
the  product,  than  there  are  decimal  places  in  the  mwltipiicand  and  multiplier 
counted  together;  then  all  tlie  figures  to  the  left  hand  of  the  point,  will  be 
dollars,  and  those  to  the  right  hand,  dimes,  cents,  mills.  Sec.  which  will  be  the 
interest  required. 

Should  there  be  a  remainder  in  taking  one  sixth  of  the  days,  reduce  it  to 
a  vulgar  fraction,  for  which  take  aliquot  parts  otthe  multiplicand.     Thus, 
If  the  remainder  be  Izz:',)  divide  the  multiplicand  by  6 

If 2=ii by  3 

If 3=i by  2 

If 4=1 by  3  twice; 

If 5  :^J  and  I by  2  and  3 

The  quotients  which  in  this  way  occur,  must  be  added  to  the  product  of  the 
principal  niuliip'icd  by  half  the  months,  Sec.  the  sum  thus  produced,  will 
Ije  the  interest  required. 

When  there  are  days,  but  a  less  number  than  6,  so  that  6  cannot  be  contained 
in  them,  put  a  cypher  in  place  of  the  decimal  at  the  right  hand  of  the  months, 
then  proceed  in  all  respects  as  above  directed. 

Note.  Jn  .casting  interest,  each  month  is  reckoned  30  days. 

EXAMPLES. 
1.  What  is  the  interest  of  Dolls.76,5A  for  1  year,  7  months,  and  1 1  days  ? 

The  number  of  months  being  19,  the 
greatest  even  number  is  18,  half  of  which  is 
9,  which  I  write  down  ;  then  seeking  hovr 
often  6  is  contained  in  41,  (the  sum  of  the 
days  in  tUe  odd  month  and  the  given  days) 
I  find  it  will  be  6  times,  which  I  also  set 
down  ut  the  right  hand  of  half  the  even 
number  of  months  Ti-r  a  decimal,  by  which 
Ans.  r,  4    1    1   6  2  together  I  multiply  the  principal.    In  taking 

one  sixth  of  the  days  (41)  there  will  be  a  re- 
mainder of  5z=j  and  ;^<for  which  I  take,  first 
one  half  the  multiplicand,  that  is,  divide  the 
multiplieand  by  2,  then  by  3  and  these  quo- 
tients added,  with  the  products  of  half  the 
even  number  of  months  cJ'c.  the  sum  of  them  will  shew  the  interest  required, 
observing  to  count  of^^7^'0  more  figures  for  decimals  in  the  product  than  there 
are  decimal  figures  in  both  the  multiplier  and  muhipiicand,  counted  together. 

For  tlie  conciseness  and  simplicity  of  the  above  METHOD,'it  is  conceived,  that 
Instructors  will  recommend  it  to  their  Pupils  in  preference  to  any  other. 


OPERATION 

7 

6; 

»5 
9 

4 
6 

4 

5 

9 

2 

4 

6   8 

8 

8 

6 

3 

8 

2 

7 

2 

5 

5 

1 

7,4 

1 

1 

6 

2 

\,y-Tsj 

S 

Sect.  II.  4. 


SIMPLE  INTEREST. 


101 


2.  What  is  the  interest  of  Dots. 
5,93  for  2  years  and  8  months  ? 
Ans.  94  cents.  8  mills. 

f,  93 
16 


5S  SS 

IS 


3.  What  is  the  interest  of  DolU^ 
67,62  for  3  years  and  2  months  ; 
Ans^  12  Dolls.  84  cents  7  7nz7/«. 


/9 


60  S^^ 


4.  What  is  the  interest  of  9 1 
cents  for  27  years  ? 

AnSi,   1  doL  4:7cta.  Am. 

ST 


18% 


When  the  interest  on  any  sum  is  requir- 
ed for  a  great  number  of  years,  it  will  be 
easier,  first  to  find  the  interest  for  1  year, 
then  multiply  the  interest  so  found  by  the 
number  of  years. 

5.  What  is  the  interest  of  Bollt. 
2870,32  for  10  days  ? 
Ans.  4  dolls',  78  cts.  3m. 


frit /?z 


92, 


When  the  rate  is  any  other  than  6  feu  cent*,  Jirstjind  the  interest  at  6  fier 
cent,  then  divide  the  interest  so  found  by  such/iarts  as  the  interest  at  the  rate  re- 
(juired^  ey^cceds  or  falls  short  of  the  interest  at  6  fier  cent,  anil  the  quotient  added 
to  or  subtracted  from  the  interest  at  6  fier  cejit^  as  the  case  lyiay  be,  will  give  the  , 
intereat  at  the  rate  required,      --  |. 


6.  What  is  the  interest  of  Dolls. 
137,84  for  2  years  and  6  months  at  i 
fier  cent  ?  Am.  DolU.'^l  7.23 


68i  zo 

xrMAML 


7.  What  is  the  interest  of  D^^. 
79\q7  fovMq  months  at  8  /je<  csntlT- 
^jM'.  Dolls.  5j27\. 


JO 


1C2 


SIMPLE  INTEREST. 


SCET.    II.    4. 


8.  What  is  the  interest  of  DolU. 
2,29  for  1  month,  19  days  at  3 /ier 
cent  ?  Ann.  9  millfi, 

10.  What  is  the  interest  of  Dolls. 
1600  for  one  year,  and  3  months  ? 
Ans.    120  dolls. 

12.  What  is  the  interest  of  Dolls^ 
ir,68  for  1 1  months,  and  28  days  ? 
AA,n8,  Dolls.  1,0 5 -4 


14.  What  is  the  interest  of  105; 
1  year,  7  months,  and  6  days  ? 

Ans.  \0  dolls,  \octs.  8m. 


61 


9.  What  is  the  interest  of  Dolls. 
1 8  for  2  years,  14  days  at  7 per  cent  ? 
Ans.  2  dolls.  5  6  cts.  9m. 

1 1.  What  is  the  interest  of  Dolls. 
5,8 1 1  for  one  year  and  1 1  months  ? 
Ans.  66  cents  8%. 

13.  What  is  the  interest  of  Dolh. 
861,12  for  9  months,  25  days  at  7 
fier  cent  ?         Ans.  49,394. 

1  5.  What  is  the  interest  of  Dolls. 
85  for  9  months  ?       Ans.  dolls.  3,87., 


16.  What  is  the  interest  ol  78  Dolh< 
36cts.  for  5  years  10  months,  and  3 
•lays  ?       Ans.  27  dolls.  46  cCs.  5m. 


17.  Wl)at  is  the  interest  of  812 
Dolls.  oOcts.  for  2  years,  8  months, 
and  4  days  ? 

Ans.  130  dolls.  SOcta.  9m. 


To  this  mode  oi"  computing  interest,  I  would  add  from  the  "  Massachusetts 
Justice -j"  a 

METHOD. 

Of  com[mting  the  interest  due  on  bonds,  notes  ^  bV.  ivhcn  partial  pay- 
mems  mav  at  different  tiyncs  be  made^  as  established  by  the  Courts 
6f  Lain  in  Massachusetts, 

RULE. 

Cast  the  interest  up  to  the  first  payment,  and  if  the  payment  exceed  the  in- 
terest, deduct  the  excess  from  the  principal,  and  cast  the  interest  upon  ihe 
remainder  to  the  time  of  the  second  payment.  If  the  payment  bs  less  than  the 
interest,  place  it  by  itself,  and  cast  on  the  interest  to  the  time  of  the  next  pay- 
ment, and  S0  on,  until  the  payments  exceed  the  interest,  then  deduct  the  ex- 
«ess  from  the  principal,  and  proceed  as  before. 

EXAMPLES. 

Suppose  A  should  have  a  bond  against  B  for  1 166  dollars,  66  cents,  and  6 
mills,  dated  May  1,1796,  upon  which  the  following  payment  should  be 
made,  viz. 

1.  December  25,   ir?6     -     -     - 

2.  July  10,   1797, 

3.  September  1,  1793,  -  -.  -  ■ 

4.  June  14,  1799, 

5.  Apiil  15,  18G0,     -     -     -     -     - 
What  will  be  due  upon  it  August 


Dollars,  Mills^ 

-      166,   666 

-     -      16,   666 

-     -    50,  000 

-  333,  333 

620,  000 


Months^  DaySi 


1301 


7 

6 

13 

9 

10 
15 


24 
15 
21 
13 

18 


Ail's.  Dolls.  237  96  cents. 


To  facilitate  the  operation  let  the  space  of  time  from  the  date  of  the  Bond 
to  the  day  of  the  first  payment,  and  from  the  time  of  one  payment  to  that  of 
another,  and  from  that  of  the  last  payment  to  the  time  of  settlement,  be  first 
•omputed  and  set  down  against  the  day  of  pE^yment  as  above.     Then  set  down 


S-Ecr.  11.  4. 


SIMPLE  INTEREST. 


the  sum  on  which  the  interest  is  to  be  cast,   with  the  interest  and  payments 
in  columns  thus, 


Principal . 

Dolls.  Milh. 
1166,666 
12  1,167 

1045,499 
1045,499 
1045,499 

245,093 

800,406 
579,847 


Time. 


Mo.  Da. 
7     24 


6      15 

13      21 
9      13 


10      1 


220,559       I        15      18 


Interest. 


Doll.<i.  M. 
25,490 


33,978 
71,616 
49,312 

154,906 

40,153 

17,203- 


Payments. 


Tlie  last  remainder 

Interest  from  the  last  p?iyment 


Dolh.  M. 
166,606 


16,666 

50,000 

333,333 

399,999 

620,000 


220,559 
17,203 

237,762 


Excess. 


Dolh.M. 
121,167 


245,095 
579,847 


Sum  due  Aug.  1st.  1801. 

2.  Supposing  a  note  of  867  dollars,  33  cents  dated  ^an.  6,  1794,  upon  whicli 
the  following  payments  should  be  made,  viz. 

Dolls,  cts. 

1.  April  16,   1797,     -     -     -     -   136,44 

2.  April  16,   1799,     -     -      -       319, 

3.  Jan.    1,    1800.     -      -     -     -     518,68 
>Vh^t  Avouldbe  due  July  11,  1801  ?        Jns.  Dolls.  215,103. 


s-zo  3  ^8 

iZ0398 

y  ^ 


^5^. 


dJ6y 


^— A/dPd?^. 


^^//  A 


/<£>  o  y,  /jj. u^/f]t. u.i>K 


104  COMPOUND  INTEREST.  Sect.  II.  4. 

COMPOUND  INTEREST, 


Is  calculated  by  adding  the  interest  to  the  principal  at  the  end  of  each  year 
and  inakini;  the  amount  the  principal  for  the  succeeding  year  ;  then  the  given 
principal  subtracted  from  the  last  amount  the  remainder  will  be  the  compound 

interest. 

A  concise  and  easy  Method  of  casting  Compound  Interest ^  at  6 
per  cent,  on  any  sum  in  Federal  Money, 

RULE. 
Multiply  the  given  sum,  if 

ror  2  years^  by  112,36  For  7  years,  by  150,3630 

3  years —  119,1016  8  years —  159,3848 

■ 4  years —  126,2476  9  years —  168,9478 

5  years —  133,8225  10  years —  179,0847 

6  years —  141,8519  U  years —  189,8298 

Note.  1.  Three  of  the  first  highest  decimals,  in  the  above  numbers, 
will  be  sufficiently  accurate  for  most  operations  ;  the  product,  remember- 
ing to  move  the  separatrix^wo  figures  from  its  natural  place  towards  the 
Jefi  hand,  will  then  shew  the  amount  of  principal  and  compound  interest 
for  the  given  number  of  years.  Subtract  the  principal  from  the  amoutit 
and  it  will  shew  the  compound  interest. 

2.  When  there  are  months  and  days  ;  first,  find  the  amount  of  principal  and 
compound  interest  for  the  years,  agreeable  to  the  foregoing  method,  then, 
for  the  months  and  days  cast  tlie  simple  interest  on  the  amount  thus  found  ; 
this  added  to  the  amount  will  give  the  answer. 

3.  Any  sum  of  money  at  Compound  Interest,  will  double  itself  in  11  years, 
10  months,  and  22  days. 

i 

EXAMPLES. 

1.  What  is  the  compound  in-  2.  What  is  the  amount  of  1^236  at 

lerest  of  $56  75  for  11  years?  compound   interest,   for  4  years,   7 


months  an4  6  days 


OPERATION.  OPERATIOHr, 

5  6,  75       ^  12  6,  2476 

18  9,  829  236 


51075  7574856 

11350  3787428 

45400  2  524952 

6  10  7  5  


5400  ^29   7,  944336  Jmmint 

6   7   5  3,  6  [for  4  years. 


glO  7,  7279575  Jjnount.  17   8  7   6  6  4 

5   6,7   5  Principal  subtracted.  8   9   3   8   3   2 


{mo.  6  daijs. 


^5  Oj  9  7  CompQund  interest.  ^10,  725984  Interest  for  7 

2  9    7,  9  4  4  Amount  for  4  years 


S  3  0   8,  6  6  9         Jnsiver. 


[added. 


Sect.  II.  4.       SUPPLEMENT  lo  S.  INTEREST.         105 

Supplement  to  ^^.  ^nterc^t. 

QUESTIONS. 

1.  IVHAfh  interest  ? 

2.  JVhjT  is  understood  bxj  6  P£/?  cENr  ?  3  J*£ff  cEN-r?  8  P£fl  cr^r,  fJJ'c. 

3.  WHAffier  cent.fier  annum  is  alloived  by  Law  to  the  Lender  for  the  use  cf 

his  Money  ? 

4.  WHAfis  understood  by  the  Principal  ?  the  RAfE  ?  the  AMOUNf  ? 

5.  Of  how  many  kinds  isinteres/  ?  in  what  does  the  difference  consist  ? 

6.  Hoif/-  is  simfile interest  calculated Jor  one  year  in  Federal  Money  ? 

7 .  For  more  years  than  one^  how  is  the  interest  found  ? 

8.  Wheh  there  are  months  and  days  what  is  the  method  of  fir  oce  dure  ? 

9.  What  other  MEtHOD  is  there  oj  casting  interest  on  sums  in  Federal  Money  ? 

10.  When  the  days  are  a  less  number  than  6,   so  that  6  cannot  be   contained  in 

themj  what  is  to  be  done  ? 

1 1.  How  is  simfiU  interest  cavt  in  pounds,  shillings,  /lence,  and  farthings  ? 

12.  When  fiartial  fiayments  are  made,  at  different  times,  how  is  the  interest  caU 

ciliated  ? 

EXERCISES. 

1.  What  is  tlie  interest  of  91  eZ^o//*.   2.  What  is  the  interest  of  93  Dolls, 
72  cts.  for  1  year  and  4  month's  ?  17  ct3.  \  1  clays  ? 

Ans.  DoU».7j^o^7.  Ans.  17  cts. 


93  /  r . 

3.  What  is  the  interest  of  Dolls.  5,19,        4.  What  is  the  interest  of  Dolh. 
for  7  months  ?  1,07  for  3  years  6  months,  and  IS 

jitis.  IS  cts.  \m.  days  ?         jins  22  cts.  7m. 

l_ 

I  S,I6 


O 


l,OT 

2/^ 

12/3  ^r 


IQO i  SUPPLEMENT  to  S.  PNTEREST.     Sect.  IL  4. 

5.  Whatis  the  ihter&stof^  41   lis.         6.  What  is  the  interest  of  Dolls. 
3|d.  for  a  year  and  2  raonths  ?  273,51  at  7  fier  cent  for  1  year  and- 

Ans.^l   18  2i  10  days?         jim.  JDolls,.  19^677. 

%r:>.  r/ 


^/.  II. 

3^ 

2J90./8.I 

/    to 

0% 

IS/IS 

Z/t6 

/£^l£>6 
2.  r3  57 

t.  ScpfoSiKG  a  note  of  i)o&.  3 1 /,92 dated  July  S',  lYsr,  on  which  \*ere 
the  following  payments,  Sept.  1 3,  1799,  BolU.  208,04.  March  10,1800  Ddlu 
76  ;  whatwas  the  sum  due  Jan.  1,  1801  \        Jina.  Dolla.  85,991, 


62cr.  11.  5.    COMPOUND  MULTIPLICATION, 


ao7 


§  5,  Compound  Mttlti0mtmh 

Co^irouND  Multiplication  is  when  the  multiplicand  consists  of  sever- 
al denominations.     It  is  paniculufly  Aisefui.in  finding  the  value  of  Goods. 

The  different  denominatrions  in  wh^t  was  formerly  called  Lnnv/ul  Money, 
•render  this  rule  with  some  others  in  Arithmetic,  as  Comfiound  Division  and 
/-'rac/'/c<?,  rules  of  great  usefulness,  quite  tedious,  and  the  variety  of  cases 
necessarily  introduced,  extremely  burthensome  to  the  memory.  This  iu7n- 
herofthe,  mind  might  be  almost  wholly  dispensed  with,  were  the  -habit  of 
reckoning  in  Federal  Money  generally  adopted   thro*  the  U.  States. 

Foil  important  reasons,  Pounds^  Shillings^  Tence^  and  farthingu  ought  to 
fall  wholly  into  disuse  :  Federal  M<=>ney  is  our  J^ational  Currency ;  the  Schol- 
lar  might  encompass  the  most  uscfal  rules  of  Arithmetic  in  half  the  time  ; 
the  value  of  commodities  bought  and  sold,  might  be  cast  with  half  the  troub- 
le, and  with  much  les«  liability  to  errors,  were  all  the  calculations  in  money 
universally  made  in  Dollars^  Cenr.Sy.and  Mills.  But  this,  to  be  practised  must 
'^bc  taugiit ;  it  must  be  taught  in  our  Schools, and  so  long  as  the  prices  of  goods, 
and  almost  every  ma.V's  accounts  are  in  Pounds^  Shillings^  Penccy  a?id  Far- 
t/iingSf  this  mode  of  reckoning  must;not  be  left  untaught. 

To  comprise  the  greater  usefulness,  and  also  to  shew  the  great  advanta^ 
which  is  gained  by  reckoning  in  Federal  Money,  I  have  contrasted  the  two 
^odesof  account,  and  in  separate  columns,  on  the  same  page,  have  put  the 
same  questions  in  Old  JLaitful  and  ^n  Federfll  Money. 


<iPEIlATI0»N. 


Pounds,  S'hil.  Pence,  Farthings. 

CASE     1, 

IVh^s  the  gua?ility  does  not  exceed 
12  yards,  pounda,  k^fc.  set  down  the 
price  of  1  yardorpound^and  place  the 
quantity  underneath  the  lowest  de- 
nomination for  a  multiplier.  Begin  by 
•multiplying  the  lowest  denomination 
and  carry  by  the  same  rules  from  one 
.denomination  to  another,  as  in  Com- 
pound Addition 

EXAMPLES. 

t.  What  will  7  yards  of  qlotbcost 
»t  9/5  per  yard  ?         * 

OPERAT.tdt^. 

O    -9     Sftrice  of'\  yard. 
"    '7  yards. 


^n«.  3     5'     ilfiHce  o/'9  yards. 


:BdUany  CeritSi  Milk, 

IN    ALL    CASES. 

.Muf^TiPUY  the  pripe  and  the  quan- 
tity together  according  to  the  rules 
of  mulliplicatiqn  xwCcciuiqI Fractions^ 
and  Federal  Money,  and  the  product 
will  be  the  answer,  That  is, 

Multiply  as  in  simple  multiplica- 
tion, and  from  Xh^  prodiict  point  off 
so  many  places  for  cents  and  mills  as 
there  are  places  of  cents  and  mills  in 
the, price. 

^EXAMPLES. 

^1.  What  will  7  yards  of  cloth  co^ 
at  Sl,57  (equal  to  9/5  J  per  yard  ? 

OPERATION' 

D.    eta.  As  there  arc 

1,     57  price.         two     decimal 

7  quantity,  places  in  .the 

— price     so      ^ 

An9. 1 0,90Jirice  oj  0yds.  make  two  in 
ihe'product. 


108 


COMPOUND  MULTIPLICATION.     Sect.  II.  5, 


Founds,  ShilL  Pence,  Farthings. 

2.  What  will   9  pounds  of  sugar 
cosl  at  \0d.  per  pound  ?    Jus.  7;6. 


3.  What  will  6  yard  of.  cloth  cost 
at^l.  106-.  5d.  per  yard  ? 

Jns.  £9  2s.  Od. 


CASE    2. 

Jllien  the  qiianlily  exceeds  \2andis 
any  7tuivber  ivithin  the  Multiplicaihn 
Table.,  multiply  by  two  such  numbers, 
«s  when  multiplied  together,  will  pro- 
duce the  given  quantity. 

If  no  two  numbers  will  do  this  ex- 
actly, multiply  by  two  such  numbers, 
as  come  the  nearest  to  it,  and  by  the 
deficiency  or  excess,  multiply  the 
multiplicand,  and  this  product  added 
to,  or  subtracted  from  the  first  pro- 
duct, as  the  case  may  require,  gives 
the  answer. 

EXAMPLES. 

1.  What  wilV42  yards  of  cloth  cost 
at  1579  per  yard  ? 

OPERATION. 

^.     s.     d. 

O      15     9  price  of  I  yard, 
multiplied  by  6 


Multiplied  by 


6  firiceof&  yards. 
7 


53        1      6firiceof^2yards. 
Because  6  times  7  is  42, 1  multiply 
the  price  of  1  yard  by  6  and  this  pro- 
duct bv  7,  as  the  ruhj  directs. 


Dollars,  Cents,  Mills, 
2.  What    will    9    pounds  of  sugar 
cost  at  go,  139  per  pound  ? 

Ans.  S  1,2  51. 


3.  ^V'ilat  Will  6  yards  of  cloth  cost 
at  S5,07  per  yard  ? 

Ans.  g30,43. 


4.  What  will  42  yards  of  cloth  cost 
at  S2)625  per  yard  ? 

OPERATION. 

D.  cl».  m. 
2,  6   2    5 

4  2 


5  2   5  0 
10   5    0  0 


Si    1  0,2  5  0    Amzver. 


Sect.  IL  5.     COMPOUND  MULTIPLICATION.        109 


Pounds,  Shil.  Pence.  Farthings, 
2.  What  will  125  yards  ot  cloth  cost 
at  5/r  per  yar4?v^Ai*.  ^34.^7   W 


S.  What  will  5  I    pounds  of  tea  cost 
at  3/5  per  lb.  Jn^.  £^    IS.y.  60^. 


4.  What  will  130  yards   of  cloth 
cost  at  ^2  3s.  9d.  per  yard  ? 

.ins,  ;C284  7s.  6d. 


CASE.  O. 
When  the  mule/ilier  that  is^  the  quan- 
tity, exceeds  144,  multiply  Srst  by  10 
and  this  product  again  by  10  wliich 
will  give  the  price  of  100  yards,  &c. 
and  if  the  quaniily  be  even  hnndreds, 
iTiuUiply  the  price  of  one  hundred  by 
tlifc  nuiviber  of  hundreds  in  the  ques- 
tion, and  the  product  will  be  the  an- 
swer ;  if  there  are  odd  numbers, 
multiply  the  price  often  by  the  num- 
ber of  tens,  and  the  price  of  unity  or 

I  by  the  number  of  units,  then   these 
several  products  added  together  will 

be  the  answer. 


Dollars,  Cents,  Mills, 

5.  What   will    125   yards  of  clotli 
co&t  at  93  cents  per  yard  ? 

Ms.  Si  1 6325      "    , 


6.  What  will  51  pounds  of  lea  coU 
at  §0,583  per  lb  ? 

.'//re.  §29,733 


7.  What  will   130  yards  of  cloth 
CQ^tatg7,25  per  yard  ? 
^  Ana.  S  942,50 


no       COMPOUND  MULTIPLICATION.      Sect.  IL  §, 


Founds,  ShilL  Pen  a.  Farthings^ 

1.  What  wjll  563   yards'  of  cloth 
cost  at  C\  6'-  7f/.  per  yard  ? 
/J.  *.  d. 

1     6     7  firice  of  \  yard. 
10 


13     5    \0  firice  of  \0  yds. 
10 


4  f;rice  of  100  yds. 
5 


664   418  firice  of  500  yds. 
€>litnes  1 0yd. 70    J  5  O  firice  (f  60  yds. 
2,  limes  1  yr/i  ^    19  -9  firice  of  3  yds. 

Jins.n^   6   5firiccof56^yds. 

2.  What   will   328  yards  of  cloth  j 
cost  at  lOs.  6^-d.  yer  yard  ? 

►<i^?:*,  ^'172    173.  8d. 


Dol/ars,  Cents,  Mills, 

S.    What    will  563  yards  of  clotk 
cost-  at  •.§4,43  |>er  yard  ? 

OPERATION. 

Yards.   5   6  3 

^  4,  4  3  i 

16  8  9 
$   2   5  2 
"5252 


jg  2   4  9  4,  0  9  ^n#. 


5.  What  will  S28  yards  ofdot^ 
cost  at  S  1)757  per  yard  ? 

4ns.  %  576,295 


3.  WhiU  will  G24   yards   of  cloth 
cout  at  12s.  ^d.  per  yard  ? 

J7is,  £295  4s. 


10.  What  will  6^4  yards  of  clotk 
cost  at  $2,111  per  yard  ? 

Jns.  S  1317,264, 


Stct.ItS.  SUPPLEMENT  TO  C. MULTIPLICATION.  iU, 


Supplement  to  CompOUttb  JiBultipltCatlOn^ 

QUESTIONS. 

1.  What  ia  dqinjiound  Multiplication  ? 

2.  What  is  its  use  ? 

3-.  jIre  ofieratiorii  nioiteaiy  in  Old  LawPvI^  6r  in  FkotRAL  MdNsr. 

4.  WHAtistheruleforcomfioundMuliililication? 

5 .  TVhen  the  quantity  that  is  the  Multifiliery  excet  ds  1 2  and  is  within  the  Mul* 

ti/ilic<ftion  T((.bte  what  are  the  ste/is  to  be  taken  ? 

6.  When  two  numbers  multifilied  together  will  produce  the  given  quantity^ 

what  then  is  to  be  done  ? 

7.  When  the  multiplier  exceeds  144,  what  is  the  method  of  procedure  ? 

8.  When  the  price  of  goods  are  given  in  Federal  Money  ^  wfkit  is  the  generjU 

and  universal  rule  for  Ending  their  value  by  muUiplicati@n  ? 

EXERCISES, 

1.  A  MAN  has  38  silver  cups,  each  2.  If  a  wiati  travel  34  miles,  3  fur- 
one  weighing  I  ©z.  Spwts.  1 6grs.  fiow  longs,  and  1 7  rods  in  one  day,  how  far 
jiftuch  silver  do  they  all  contain  ?  T»ill  he  travel  in  62  days  ? 

jins.  Sib.  8oz.  I9pwts,  85T*,  jins.  2134  miles,  4/ur*  \4rods. 


3.  What  will  235  yards  of  cloth 
come  to  at^f  1  25.  5\d.  per  yard  I 
-^«».;C263  Us.^^d. 


4.  If  a  horse  run  a  mile  im  12  min» 
utes,  16  seconds,  in  what  lime  MTouKi 
he  go  176  miles  ; 

An:  I  D.  llh.  Sim.  SB*. 


}}2 


COMPOUND  DIVISION. 


Sect.  IJ.  6. 


§  6,  Compounti  vi^ibi^ion. 

Compound  Division  is  the  dividing^  of  diffevent  denominations. 

OPERATIONS. 


Founds,  Shlll.  Pence,  FanJjings, 

CASE    1. 

1.  When  the  divisor^  that  ia^,  the 
(juintity  doeti  not  exceed  12,  be§;inat 
ilie  h.igljest  denomination,  and  in  the 
masner  of  short  division,  find  how 
many  times  th«  divisor  is  contained 
in  it,  place  the  quotient  under  itspwn 
denomination,  and  it"  any  ihin^  re- 
main, reduce  it  to  the  next  less  de- 
nomination, and  divide  as  before  ;  so 
praceed  through  all  the  denomina- 
lio'ns.      ' 

2.  If  the  quantitxj  exceed  12,  and 
there  be  any  two  numbers^  which  mul- 
t? filled  together  will  firoduce  it^  divide 
the  price  first  by  one  of  those  num- 
l>ers,  and  this  quotient  by  the  other. 


6il. 


EXA^TPLES. 
.  If  5  yards  of  cloth  cost  ^'3 
what  is  that  per  pard  ? 


13s. 


OPERATION. 

5)3      13     &  firice  of  5  yards. 


0      U     Q^  firice  of  I  yard. 

FiNDiTCG  I  cannot  have  the  divisor 
(5)  in  the  first  d*:nominaiion  (^.3)  I 
reduce  it  to  shillinc^s,  (GO)  and  add  in 
the  13  shilliiis^s,  which  make  73  shil- 
lings, in  wiiich  the  divisor  (5)  is  con- 
tained 14  limes,  and  3  remain  ;  I  set 
down  the  14  Sc  the  remainder  )3  shil- 
linijs)  reduced  to  pence  (36)  and  the 
€d.  added  make  42  pence  in  which 
the  divisor  is  contained  8  times  and 
iwo  remain  ;  I  set  down  the  8,  and  re- 
duce the  two  pence  to  farthings  (8) 
in  which  I  have  the  divisor  ance{\qr. 
or  |f/.)  and  a  remainder  of -5  of  a  far- 
vhing,  which  being;  of  small  value  is 
iitgkcted. 

2.  If  48  yards  of  cloth  cost  £4  I65. 


Dollars,  Cents,  Mills. 


IN    ALL    CASES. 

Divide  the  price  by  the  quantity, 
and  point  off  so  many  places  tbr  cents 
and  mills  in  the  product  as  there  are 
places  of  cents  and  mills  in  the  divi- 
dend. 

If  the  quantity  be  a  com/iosite  num- 
bery  that  is  produced  by  the  multi- 
plication of  two  numbers,  the  opera- 
tion may  be  varied  by  dividing  the 
price  first  by  one  of  those  two  num- 
bers and  this  quotient  by  the  other. 

EXAMPLES. 

1.  If  5  yards  of  cloth  cost  Dolls.  12, 
25,  what  is  that  per  yard  ? 


n. 

5)1 


OPERATION. 

Cts. 
2,2  5 


There  are  twt> 

decimal  places  in 

.^ns.  2,  4  5  the  di\idend.     I, 

therefore, jxjint  oit 
two  places  for  decimals,  or  cents  in 
the  quotient. 


2.  If  48  yards  of  cloth  cost  Dotls, 
16,06  what  is  that  per  yard  ? 

.4ns,  Dolls.  Oyoo, 


Aid. 


what  is  that  per  pavd  ? 


Sect.  II.  6.  COMPOUND  DIVISION.  115 

Pounds,  ShiU.Peiicey  Far  things  y  Dollars^  Cents  ^  Mills. 

3.  If  24lb.  of  tea  cost  £.2  7s,  9^d.         3.  If  24  lb.  of  tea  cost  Dolls,  7,97 
what  is  that  per  lb.  ?  what  is  that  per  lb.  ? 

Jas./C.O   Is.   lUd.  Ans.  Dolla,  0,2Z2 


4.  If  35   yards  of  cloth  cost  £A2 
is.  7\d,  what  is  that  per  yard  ? 


CASE    2. 

1.  "  Having  the  firice  of  an  hund- 
red iveight  ( 1 1  '214).)  tojind the  firicd  of 
I  lb.  divide  the  given  price  by  8,  that 
quotient  by  7,  and  this  quotient  by  2, 
and  the  last  quotient  will  be  the  price 
of  1  lb.  required." 

2.  If  the  number  of  hundredweight 
be  vvQre  than  one,  first  divide  the  whole 
price  by  the  number  of  hundreds, 
then  proceed  as  before. 

EXAMPLES. 
1.  If  1  cwt.  of  sugar  cost  £S  7*.  6d. 
what  is  that  per  lb.  ? 

OPERATION. 
£.     s.    d.    q. 
8)3     7     6         firice  1  av)t. 


7)0  8  5  1  firice  of  \  Alb.  or  I  civt, 
2)  12  1  firice  of 'Xlb.  or  ^^ctut. 
Ann.  0     7      1  firiee  of  Mb, 


4.  If  35  yards  of  cloth  cost  JDolU 
141,103  what  is  that  per  yard  ? 
Ana.  Dolls.  4,031 


The  same  may  be  done  ia  Federal 
Money. 


5.  Ir  1  cwt.  of  sugar  cost  Dolls  1 1, 
%5  cts.  what  is  that  per  lb.  ? 

Ana,  10  cents. 


114 


COMPOUND  DIVISION. 


Sect.  II.  6, 


Pounds^  ShilL  Pence ^  Farthings, 

2.  If  8  cwt.  of  cocoa  cost  jCl5  *ts. 
4d.  what  is  that  per  lb.  ?     Jns,  4id. 


3.  I?  3  c-wfr.  of  sugar  cost  ^15  13*. 
Vhat  is  that  per  lb  ?  Jns.  1  Id, 


Dollars,  Cents,  Mills. 

6.  If  8  cwt.  of  cocoa  cost  jg5 1,223 
what  is  that  per  lb  ? 

^ns,   5  cents,  7  mills. 


7.  If  5  cwt.  of  sugar  cost  ^52,167 
what  is  that  per  lb  ? 

jfns.   15  cents,  5  mills. 


CASE.    3. 

5<  WffSN  the  divisor  is  such  a  number 
<25  cannot  be  produced  by  the  multi/ili- 
cation  of  small  numbers,  divide  after 
the  manner  of  long  division,  setting 
down  the  work  of  dividing  and  re- 
?H»eing." 


Sect.  II.  6. 


COMPOUND  DIVISION. 


115 


Founds ^ShilL  Pence y  Farthings. 

EXAMPLES. 
1.  If  46  yards   of  cloth  cost  aC.53 
10*.  6d.  what  is  that  per  yard  ? 


Z  ^Z^Ana, 


OPERATIOJ^r. 

/;.    s.  d.  £,  s.    d. 
46)53   10   6(1 
46 

7 

20 

46)150(5 
138 

12 

12 

46)150(3 
158 

12 
4 

46)4i(l 
46 


2.  If  263  bushel  of  wheat  cost  <C86 
7s.  lod.  what  is  that  per  bushel  ? 


3.  If  C70  f;:»llonsofwinc  cost  >C147 
!<•    1  Id.  what  ii  that  per  ^^«lloi»  ? 

^//A'.  4.S.  4j.«/. 


Doilans,  Cents,  Mills. 

8.  If  46  yards  of  cloth  cost  gl^d 
,4 1 6  what  is  that  per  yard  ? 

Am.  g3,878. 


9.  If  263  bushels  of  wheat  coBt 
g28f  j973,  what  is  that  per  bushel  ? 
An8,  21,093 


10.   If   670  gallons  of  wine  cott 
S490,32  ;  what  is  that  per  gallon  ? 
Arnt,  ^0,7  3. 


116  SvPFLEMENT  10  COMPOUND  DIVISION.  Sect.1I.  6. 


Supplement  ToCompOUnD^Ditjij^lOn. 

— — "iWli^  -:;=•  ■i'.i  -Jl'c  ^  •*'/:•  •5!?>  "Jlfr  <■■*■—— 

QUESTIONS. 

1.  IVHAf  is  Comfiound  Division  ? 

2.  When  the  firice  of  any  quantity^  Hot  exceeding  12,  ofyards^  Jiounds^  Isfc.  ia 
given  in  fioundt,  shillings ^fience  Isf  Jarthings^  honv  ia  the  firice  of  one  yardfuund  ? 

3.  When  the  quantity  is  such  a  number  as  cannot  be  firoduced  by  the  multi' 
plication  of  small  numbers^  what  is  the  method  of  procedure  ? 

4.  Having  the  firice  of  an  hundred  weight  given^  in  what  way  is  found  the 
price  of  Mb.? 

5.  If  there  be  several  hundred  weighty  what  are  the  stefis  of  ofierating  ? 

6.  When  the  price  is  given  in  Federal  Money  ^  what  is  the  method  of  operating  ? 

EXERCISES. 


Founds,  ShilL  Pence,  Far  things. 

1.  Ir  10  sheep  costyC4  5«.  7rf.  what 
is  the  price  of«ach  ?        Jns.  6/61. 


Dollars,  Cents,  Mills. 

Lfet  the  scholar  reduce  the  price  of 
the  sheep  and  of  the  cows  to  Federal 
Money,  and  perform  the  operations 
in  Dolls.  Cents  and  Mills. 


Price  of  \  sheep,  %\,A,%6. 


2.  If  84  cows  cost  £252  13s.  what 
js  the  price  of  each  ? 

jins.^2  0  Aid. 


Price  cf\  a>ty,S  10,06- 


Sect.  II.  6.  Supplement  to  COMPOUND  DIVISION;  11 


3.  If  121  pieces  of  cloth  measure 
2896  yards,  \qr.  3na.  what  does  each 
piece  measure  ? 

j^ns  23  yardsy  3qr,  2na. 


4.  If  66  tea-spoons  weigh  215,  \Ooz, 
14/jwr.  what  is  the  weight  of  each  ? 
yins.  lOftwt.  ISII^-r*. 


2  cwt.  of  rice  cost  x; 2   11*. 
?    Am.^ld, 

4 


5     Ir «.   w.. 

(>\d.  what  is  that  per  lb 


6.  AT)C,.2  11*.  6|J  for  2  cwt.  of 
rice,  what  is  that  in  Federal  Money, 
and  what  is  that  per  lb  ? 

Price  of  I  Id,  3  cents,  8  mill*. 


7.  If  47  bags  of  Indigo  weigh  12 
cty/r.  ](/r.  26lb.  A,oz.  what  does  each 
weigh  ?  ^«*.    Iryr.  1/*,  12or. 


8.  If  8  horses  eat  900  bushels,  and 
1  peck  of  oats  in  1  year,  how  much 
will  each  horse  eat  per  day  ? 

Ans.  1  ficck^  \qt.  \/it.  2^ilis. 


118  Supplement  to  COMPOUND  DIVISI0N.  Sect.  II.  6. 

9.  Divide  x;297  25,  3d.  among  4  men,  6  boys,  and  give  each  man  3  times 
80  much  as  one  boy  ;  what  will  each  man  share,  and  each  boy  ? 


OPERATION. 

The  men  have  triple  £.      s.     d.    £ .     s.     d.     q. 

shares,  therefore,  multi-     18)297     2     3(16     10      1     2^zl  boi/s  share. 
ply  the  number  of  men 
(4)   by   3,  and   add  the 
number  of  boys  (6)  for 
a  divisor. 


men.    boys, 
4  and  6 
3 

12 
6 


18 

117 
108 

9 

20 


.4ns.  49     10     4     2zzi\  man* s  share. 

Proof. 

£A9    10   4  2 

4 


)182(10 
18 


1  3  the  number  of  equal     1 2 
shares  i?i  the  ivhole    — » 


rrDivisor. 


)27(1 
18 


198      1   6  ©  men*s  share. 

16   10   1  2  and 
6 


99     0  9  0  boy^a  share. 


£.197     2   3  0  added. 


9 
4 

)36(2 
36 


10.  DiviiiEjC-39   12s.  5(s;.  among  4  men,  6  women,  and  9  boys  ;  giv«  c«ch 
man  double  to  a  wx)man,  each  woman  double  to  a  boy. 

X  .  £.    s.   d. 

a  boy*s  shi 

share. 


C  \      1      5  a  boy's  share. 

Answer.  -?  2     2    \0  a  woman's  shai 

(^4     5     8  a  inan*s  share 


Sec.  II.  7.         SINGLE  RULE  of  THREE.  119 

The  Single  Rule  of  Three,  sometimes  called  the  Rule  o/Proportiox,  is 
known  by  havinj:^  ihrtc  terms  given  to  find  the  fourth . 
It  is  of  TWO  kinds,  Direct  and  Indirect^  or  Inverse. 

— Single  Rule  of  Three  Direct. 

The  Single  Rule  of  Three  Direct  teaches,  by  having  \hree  numbers  given 
to  find  a  fourth,  which  shall  bear  the  ianae  proportion  to  the  third  that  the  se- 
cond does  to  the  first..— 

It  is  evident  that  the  value,  weight,  and  measure  of  any  commodity  is  pro- 
portionate to  its  quantity,  that  the  amount  of  work,  or  consumption  is  propor- 
tionate to  the  time  ;  that  gain,  loss,  and  interest  when  the  time  is  fixed,  is 
proportionate  to  the  capital  sum  from  which  it  arises  ;  and  that  the  effect  pro- 
duced by  any  cause  is  proportionate  to  the  extent  of  that  cause. 

These  are  cases  in  direct  proportion,  and  all  others  may  be  known  to  be 
so,  when  tha  number  sought  increases  or  diminishes  along  with  the  term  from 
,which  it  is  derived.     Therefore, 

i  If  7nore  require  more^  or  less  require  less^  the  question  is  always  known  to 
'belong  to  the  Rule  of  Three  Direct.;    ' 

More  requiring  more,  is  when  the  third  term  is  greater  than  the  first  and 
requires  the  fourth  term  to  be  greater  than  the  second. 

Less  requiring'  less,  is  when  the  third  term  is  less  than  the  first,  and  re- 
quires the  fourth  term  to  be  less  than  the  second. 

Rule. 

- — ^*  1.  State  the  question  by  making  that  number  which  asks  the  question 
"  the  third  term,  or  putting  it  in  the  third  place  ;  that  whicli  is  of  the  same 
*'  name  or  quality  as  the  demand,  the  first  term,  and  that,  which  is  of  ihe  same 
*'  name  or  quality  with  the  answer  required,  the  second  term." 

"  2.  Multiply  the  second  and  third  terms  together,  divide  by  the  first, 
"  and  the  quotient  will  be  the  answer  to  the  question,  which  (as  also  the  re- 
"  mainder)  will  be  in  the  same  denomination  in  which  you  left  the  second 
"  term,  and  may  be  brought  into  any  other  denomination  required.*'   - 

The  chief  difficulty  that  occurs  in  the  Rule  of  Three,  \^  the  right  placing* 
of  the  numbers,  or  stating  of  the  question  ;  this  being  accomplished  there  i*} 
nothing  to  do,  but  to  multiply  and  divide,  and  the  work  is  done. 

To  this  end  the  nature  of  every  question  must  be  considered,  and  the  cir- 
cumstances on  wiiir;h  the  proportion  depends,  observed,  and  common  cense 
will  direct  this  if  t  lie  terms  of  the  question  be  understood. 

The  method  of  proof  is  by  inverting  the  order  of  tiic  question. 

Mjie.  I.  Ik  the  first  and  third  tenns,  bcth  or  either',  he  of  difTirent  de- 
noniinalions,  both  terms  must  be  reduced  to  the  lowest  denomination  men- 
tioned in  either,  before  stating  the  question. 

2.  If  the  second  term  consist  of  difl'erent  denominations,  It  must  be  redu- 
ced to  the  lowest  Ucnominaiion  ;  the  fourth  term,  or  answtr  will  then  be  found 
in  the  same  denomination,  and  must  be  reduced  back  again  to  the  highest  de- 
nomination possible. 

3.  After  division  if  there  he  any  remainder,  and  the  quotient  be  not  in  the 
lowest  denomination,  it  must  be  reduceil  to  the  tiext  less  denomination,  divi- 
ding aft  bcfor.c.  So  continue  to  do,  till  it  is  brought  to  the  lowest  denomina- 
tion, or  till-nothing  remains. 

4.  In  every  question  there  is  a  supposition  and  a  demand  ;  the  svTpposition 
is  implied  in  the  two  first  terms  of  the  statement,  ilie  demand  itt  the  third. 


120     SINGLE  RULE   of  THREE  DIRECT.    Sec.    II 


1 


5.  When  any  of  the  terms  are  given  in  Federal  Money  the  operation  is  con- 
ducted in  all  respects  as  in  simple  numbers,  observing  only  to  place  the  point, 
or  separatrix  between  dollars  and  cents,  and  to  point  off  the  results  according 
to  what  has  been  taught  already  in  Decimal  Fractions^  Federal  Money ^  and 
further  illustrated  in  Compound  Division. 

6.  When  any  number  of  barrels,  bales,  or  other  packages,  or  pieces  are 
given,  if  they  be  of  equal  contents,  find  the  contents  of  one  barrel  or  price,  S;c. 
in  the  lowest  denomination  mentioned,  which  multiply  by  the  number  of  pie- 
ces. Sec.  the  product  will  be  the  contents  of  the  whole.— If  the  pieces,  Sec.  be 
of  unequal  contents  find  the  content  of  each,  add  these  together,  and  the  sum 
of  them  will  be  the  whole  quantity. 

7.  The  term  which  asks  the  question,  or  that  which  implies  the  demand, 
is  generally  know  by  some  of  these  words  going  before  it ;  How  much  ?  How 
many  ?  How  long  ?  What  cost  ?  What  will  ?  Sec. 

EXAMPLES. 
1.     If  9/^5  of  tobacco  cost  6*,  what  will  25^6*  cost  ?  ♦ 

OPERATIOU. 

Ibfi. 
25  :  to  the  answer. 


lbs 
As  9 


6 

25 


30 
12 

8,  d. 

9)150)  16  8  answer, 
9 

60 

54 

6 
12 


Here  %Slbs  which  asks  the  question, 
(what  will  V-Slbs.  ^c.J  is  made  the  third 
term,  by  being  put  in  the  third  place  ; 
9lbs.  being  of  the  same  name,  the  first 
term,  and  6s.  of  the  same  name  with  the 
term  sought,  the  second  term. 

I  MuTiPLY  the  second  and  third 
terms  together  and  divide  by  the  first. 
The  remainder  (6)  I  reduce  to  pence, 
and  divide  as  before.  The  quotients 
njake  the  answer,  16/8. 


9)72(8 
72 


00 


By  inverting  the  order  ot  the  question  it  will  stand  thus, 
2     If  6s.  buy  9lbs,  of  tobacco,  wliat  will   1668  buy  ? 


6 
12 


16 
12 


72  pence  200  pence. 

pence,     lbs.    pence. 
Js,72    :     9   :   :  200 
200 


Here  the  term  which  asks  the 
question  (16s8)  is  of  different  de- 
nominations; it  must,  therefore,  be 
reduced  to  the  lowest  denomina- 
tion mentioned  (pence)  as  must 
also  the  other  term  of  the  same 
name,  consequently,  to  be  the  first 
term. 


72)1800)25  Ibs.antwar. 
144 


160 
160 


Sect.  IL  7.     SINGLE  RULE  of  THREE  DIRECT.     121 


Jgain — By  inverting  the  order  of  the  question. 

3.     If  16/8  (=200  fience)  buy  25lbs.  of  Tobacco,  how  much  will  6s(=72 
fience)  buy  ? 

OPERATION. 

d.  lbs.  d. 

As  200    :    25     :    :     72 

72 


50 
175  These  three  questions  arc  only  the 

— —  first  varied  ;  they  shew  how   any  ques- 

2100)18100(9/^5.  Ans.      tion  in  this  Rule,  may  be  inverted. 
18 

4.     If  \oz.  of  Silver  cost  6/9  what  will  be  the  price  of  a  silver  cup  that 
weighs  9oz.  Afiiut.  \6grs, 

oz.  s.     d.                                      oz.  tiwt.  grs. 

\  6       9                                            9       4  16 

20  12                                                  20 

M                As  each   of  the 

20fiivt.  %\pence.                                 I84jiwt.  terms  contains  dif- 

24  *                     24  ferent     denomina- 

-„ tions,    they    must 

80  752  all  be    reduced  to 

40  368  the  lowest  denom- 

ination  mentioned. 

480  g-w.  4432  ^r*. 

grsr        d.         grs. 
As  480  :   81    :  :  4432 
4432 

162 

243 
324 
324 


d.      q. 


480)358992(747  3  J  Amivcr,  which  must  be  reduced  to 
3360  the  highest  denomination  ;  thus, 

1  2)7  4  7      3|y. 

2299 

1920  2|0)6|2      M. 

^'3  Is   3d.  S^gr.  Ans. 


3792 
3360 

432 
4 

)  1 728(11 
1440 

288 


Q 


122     SINGLE  RULE    or   THREE  DIRECT.    Sect.  II.  7. 

5.  If  6  horses  eat  21  bushels  of  oats  in  3  weeks,  how  many  bushels  will  20 
horses  eat  in  the  same  litne  ? 

^ns.  70  biidheii.  The  same  quefitio7i  inverted. 

6.  If  20  horses  eat  70  bushels  of  oats 
in  3    weeks,   how   many  bushels  will  6 

? 

Jns.  21  bushels. 


The  statement  of  every  question  re- 
q\iires  thought  and  consideration  ; — 
here  are  Jour  numbers  given  in  the 
question  ;  to  know  which   three  are  to 

be  employed  in  the  statement  there  can  be  no  difficulty  if  the  Scholar  proceed 
,  deliberately  and  as  his  rule  directs — first,  consider  which  of  the  given  num- 
bers it  is,  that  asks  the  question  ;  that  determined  on,  put  it  in  the  third 
place,  then  seek  for  another  number  of  the  same  name,  o?M^ind,  put  that  in 
the  first  place,  the  second  place  must  now  be  occupied  by  that  number  which 
is  of  the  same  name  or  kind  with  the  number  sought  ;  when  these  steps  are 
cautiously  followed,  the  Scholar  cannot  fail  to  make  his  statement  right. 

7.  If  an  Ingot  of  silver  weigh  36oz.         8.  A  Goldsmith  sold  a  Tankard 

lOpwt.  what  is  it  wortr,  at  5.9. per  ounce  ?     for  C^O    12.9.    at  the  rate  of  5s.  4d. 

*  Jns./19  25.  6d.  per  ounce,  I  demand  the  weight  of 

it.         Jns.  3 9 or.  ISJiwt. 

t^6  ^^      it:  2«.-tf^|_j,^ 


6T6 


310 

2)Z0 


Sect.  II.  7.     SINGLE  RULE  of  THREE  DIRECT.      123 

9.  Tf  a  family  oF  10  persons  sper.d  |  10.  If  a  family  of  30  persons  spend  9 
3  busiielsof  malt  in  a  month,  how  |  bushels  of  malt  in  a  month,  how  many 
many  biisliels  will  serve  ihem  when  |  bushels  will  serve  a  family  of  10  per- 
there  are  30  in  the  family  \  sons,  the  same  time  ? 

Jtis,  9  bushels,  -/ins.  3  bu^iek. 


U.  If  12  acres,  3  roods  produce  78  quarters,  3  pecks,  how  much  will  35 
acres,  1  rood,  20  poles,  produce  ?  Ans,  216  quarters^  5  bushels,  1^  fieck, 

/Z."^:  T8.^y:  3T.J  .to 

^    z       ^ 

s/     inr     /^/ 

1_ 

■    ,4 

/90^  '    ZI6—S--li  c  '>'-■>. 


I(S?>6 


1:24      SINGLE  RULE  of  THREE  DIRECT.    Sect.  IL 


12.  If  5  acres  1  rood,  produce  26  quarters  2  bushels,  how  many  acres  will 
be  required  to  produce  47  quarters  4  bushels  ?         »^ns.  9  acres,  2  roods. 

2.6.2:  S.l::^/.^ 


±10 


1/ 


vIoJJTio 


65 
169 


5&0 


1680 

63 


As 


13.  If  565  men  consume  75  barrels 
of  provision  in  9  months,  liow  much 
wili  500  men  consume  in  the  same 
time  ?  jiri^s,  102^|  barrels* 

T50 


5-;  zro  ^  ^^ 


14.  If  500  men  consume  lO^ff 
barrels  of  provisions,  in  9  months, 
how  much  will  365  men  consume 
in  the  same  time  ? 

OPERATION. 

bar r eh* 
102f| 
Multifily  by    73   the  denominator 
— —  of  the  fraction* 
306 
714 
jidd  54  the  numerator* 


^s  500 

Note.  In  the  14th  example,  in  order  to  em- 
brace the  fraction  (4*  of  a  barrel)  the  inte- 
l^ers,  102  barrels  must  be  multiplied  by  the 
clenominutor  of  the  fraction,  (73)  and  the 
luimeravor,  (54)  added  to  the  product. 

After  division,  the  quotient  must  be  di- 
vided by  the  denominator  of  the  fraction, 
and  this  last  quotient  will  be  the  answer,  all 
which  may  be  seen  in  the  example. 

The  Scholar  must  remember  to  do  the 
same  in  all  similar  cases. 


7500  :  :   365 

7500 

182500 
2555 

5]00)27375!00 


:)5475(r5  Jns. 
511 


165 


Sect.  II.  7.    SINGLE  RULE  of  THREE  DIRECT.      125 


1-5.  If  I  give  6  Dolls,  for  the  use  of  100  Dolls,  for    12  TTJonths,  what  must 
I  give  for  Dolls.  337,^2  tVie  same  length  of  lime  ? 

OPERATION. 

Here  in  the  tliii'd  term  I  had  two 
decimal  places,  (82)  or  places  of 
cents  multiplied  by  the  second  terni 
(6j  I  point  off  two  places  for  cents 
(,92)  in  the  product,  which  divided 
by  100,  I  point  off  three  decimal  pla- 
ces in  the  quotient  equal  to  the  num- 
ber of  decimal  places  in  ^e  dividend 
(^,92  cents  and  0  annexed  to  the  remain- 
der) ihere  being  no  decimal^  in  the 
divisor. 


As 

X). 

IOC 

D. 

:6 

D  Cts. 
:  :  357,82 
6 

100' 

2146,92{21,469A 
200 

146 
100 

469 
400 

692 
600 

920 

900 

Jus 

20 


16.  How  much  land  at  ^2,50  per  acre  should  be  given  in     The  Scholar  i» 
exchange  for  360  acres,  at  S^75  per  acre  ?  |  desired  to  invert  and 

Ans*  540  acres,  \  firove  the  question. 


%Soi  360:: 


57  r 

2?60 


ZXSOO 


/oo 

/oo 


^ 


z?ooo 

io8o 


3;^^  17^^06177360. 


^v^^, 


O 


17.  If  I  buy  7lb.  of  sugar  for  75  cents, 
flow  much  can  I  buy  for  6  dollars  ? 
Jns,  56/6. 


rs- 


r: 


600 


600  , 

ys)4zooiS6  -^. 


N.  B.  Sums  in  Federal  Money 
are  of  the  same  denomination 
when  the  decimal  places  in  each 
are  equal. 
To  reduce  sums  in  federal  money  to 
the  same  denomination^  annex  so 
many  cyphers  to  that  sum  which 
has  the  least  number  of  decimal 
places,  or  places  of  cents,  mills 
&c.  as  shall  make  up  the  dcTi- 
r.icnry. 


126     SINGLE  RULE  OF  THREE  DIRECT.    Sect.  IL  7$ 

18.  If  I  buy  76  yards  of  cloth  for  |        19.  A  man   spends    g3,25  per 
§113,17  what  did  it  cost  per  Ell  Ek- |  Week,  what  is  that  per  annum  ? 
glish  ?  ^n.f.  Sl,861.  .4ns.  gl69,4G4. 


f/^ 


50^ 


lOG 


20.  Bought  a  silver  cup  weighing  9oz.AJi'ivt,  \&grs  for  iC3  3*.  3fl?.  3|y, 
what  was  that  per  ounce  ?  Ans.  6*.  9c/. 

JlfT         "JW  ^0 

11966I40 

ISXOO 
/3Z66 


Sect.  II.  7.    SINGLE  RULE  of  THREE  DIRECT.       127 


2 1 .  There  is  a  Cistern,  which  has  4 
cocks  ;  the  first  will  empty  in  10  min- 
utes ;  the  second,  in  20  minutes  ;  the 
third  in  40  minutes  ;  and  the  fourth  in 
80  minutes  ;  in  what  lime  will  all  four 
running  together  empty  it  ? 
Min.  , 

Cisi.         Min, 
1      :   :      60     : 


flO 

J  20 

•^  40 

1^80 

In  1  hour  the  4  cocks 

would  empty 
Then, 

Cist.     Min.      Cist 
As   11,25   :  60 
/ 


Cist. 

r6 

S   1,5 

L   .75 


22.  A  MAN  having  a  piece  of  land 
to  plant,  lured  two  men  and  a  boy  to 
plant  it,  one  of  the  men  could  plant 
it  in  12  days,  the  other  in  15  days, 
and  the  boy  in  27  days  ;  in  how  long 
time  whould  they  plant  it  if  tliey  all 
worked  together  ? 

Ans,  5,346  days 


'  60 


11,25   Cist. 


6d 


Min. 
:   1   :  5,33  Jns. 


JJ,K)  6000(6%'^  3  -^-^' 


//  >%x^ 


//,2^  60000  \^)3^6»^ 
58fOO 

flfto 


fTsrr 


23,  A  MERCHANT  bought  270  quin- 
tals- of  cod  fish,  for   g780  ;  freight 
^37, 70  ;   duties    and  other   charges 
§30,60  ;  what  must  he  sell  it  at,  per 
quintal  to  gain  §143  in  the  whole  ?      ] 
ylns.  §3,671 
The  sum  of  all/ he  exficncea  ofthcjish 
vjilh  the  Merchant's  gaiJi  must  be  found 
for  the  second  term. 


24.  If  a  staffs//.  8m.  in  length,  cast 
a  shadow  of  6  feet  ;  how  high  is  that 
steeple  whose  shadow  measures  153 
feet  I 


zro 


no 


nj 


Ans 

1836 

68 


\  Ail  feet 


/8S6 


4 


128    SINGLE  RULE  of  THREE  DIRECT.       Sect.  II.  7. 

25.  Bought  12  pieces  of  cloth  each  |  25.  Bought  4  pieces  of  holland 
10  yards  at  ^1,75  per  yard,  whai  came  |  each  containing  24  EUs-EnglishjCor 
they  to  ^  jin».  ^2 10  |  $9^ ;  how  much  was  that  per  yardi  ? 

^ns,  80  CenU. 


f6 

o 


27.  BotJG'HT  9  Chests  of  tea,  each  weighing  SC  S/rs'.  21/5.  atyS'4  9s.f$fr 
cv>i,  what  came  they  to  ?  jlns.  ^147   13x.  8M. 

C..,,.     ti>  ^ 

7f 

/B?> 
%o  9 


zoos 
601 


~  11%.  ^'if 


Sect.  II.  7.  SINGLE  RULE  of  THREE  DIRECT.        129 

28.  a: 

but  S607, 


28.  A  Bankrupt  owes  in  all  972  dollars,  and  his  money  and   effects  are 
S607,50  :  what  will  a  creditor  receiva  on  %\  1,333  ? 


Mi.  S7,0«S. 

6ozro 

dOt"f 

Xf/b 

1^1 


\ 


S9.  A  owes  B  iG347.'^,but  B  com-  50.  If  a  person  whose  rcHt  is  SM5, 

pounds  with  him  for  13*.   4^/.  on  the  pays  ^12, 63    of  parish     taxes,   how 

pound  ;  what  must  he  receive  for  his  much  should  a  person  pay  whose  rent 

debt?          ^;i.».>C2316   13«.  W.    ./  is  ^378.                .i/»s.  §32,925. 


^S'O  ~TiY27 

UhJTzJ^iWoJoX^-^6  00  0  ~TSW 

00(1 

R 


I 


130    SINGLE  RULE  of  THREE  INVERSE.     Sect.  IL  7; 

INVERSE  PROPORTION. 

In  some  questions  the  number  sought  becomes  less,  when  the  circum- 
stances from  which  it  is  derived  become  greater.  Thus,  when  the  price  of 
goods  increases  the  quantity  whieh  may  be  bought  for  a  given  sum  is  small- 
er. When  the  number  of  men  employed  at  work  is  increased,  the  time  in 
which  they  may  complete  it  becomes  shorter  ;  and,  when  the  activity  of  any 
cause  is  increased,  the  quantity  necessary  to  produce  any  given  effect  is  di- 
minished. 

These  and  the  like  cases  belong  to  the 

SINGLE  RULE  OF  THREE  INVERSE. 

The  Single  Rule  of  Three  Inverse  teaches,  by  having  three  numbers  giv- 
en to  find  a  fourth,  having  the  same  proportion  to  the  second,  as  the  first  has 
to  the  third. 

If  more  require  less,  or  less  require  more,  the  question  belongs  ta  the  Sin- 
gle rule  of  Three  Inverse. 

More  requiring  less,  is  when  ihQ  third  term  is  greater  than  the  first,  and 
requires  the  fourth  term  to  be  less  than  the  second. 

Less  requiring  more,  is  when  the  third  term  is  less  than  the  first,  and  re- 
quires the  fourtk  term  to  be  greater  than  the  second. 

RULE. 

"  State  and  reduce  the  terms  as  in  the  rule  of  three  direct ;  then,  multiply 
the  first  and  second  terms  together,  divide  the  product  by  the  third,  and  the 
quotient  will  be  the  answer  in  the  same  denomination  with  the  second  term.'* 

EXAMPLES. 

1.  If  48  men  cMn  build  u  wall  in  24  days,  how  many  men  can  do  the  same 
in  192  days  ? 

OrERATION. 

Men.  Days,     Men.    *       Here    the  third  term  is  greater  than  the 

As  48  :  24  :  :   192         first,  and  common  sense  teaches  the  fourth 

48  term,  or  answer  must  be  less  that  the  second, 

for  if  48  men  can   do  the  work  in  24  days, 

certainly  192  men  will  do  it  in  less  time.     In 

this  way  it  may  be  determined  if  a  question 

belong  to  the  Rule  of  Three  Inverse, 

192)1 152(6  ./fn*. 
1152 

2.  If  a  board  bq  9  inches  broad.  3.  How  many  yards  of  sarcenet, Sjr*. 
how  much  in  length  will  make  a  wide,  will  line  9  yards  of  cloth  of  Sf/r*, 
square  foot  ?  wide  ?  yins.  24  yards. 

InU.  Inh.  InB.  Inh. 

Js    12  :    12  :  ;  9  :    16  Antt. 


Sect.  II.  7.  SINGLE  RULE  of  THREE  INVERSE.       131 

4.  Lenta  friend   292  dollars  for  6         5.  A  garrison  had  provisions  for  8 

months;    sometime    afterwards,  he     months   at  the  rate  of   15  ounces  to 

lent  me  806  dollars  ;   how  long  may     each  person  per  day  ;  how  much  must 

I  keep   it  to  balance  the  favor  ?  be  allowed  per  day  in   order  that  the 

Jm.   2  months  S days,  provisions  may  last  9|^  months  ? 

Uns'.  12*1  ounces 


6.  A  garrison  of  1200  has  provis-  7.              '    How  must  the  daily 
ions  for  9  months  at  the  rate  of  14  allowance  be  in  order  that  the  pro- 
ounces   per  day,  how  long  will  the  visions  may  last  9  months  after  the 
])rovisions  last  at  the  same  allowance  garrison  is  reinforced  ? 
if  the  garrison   be  reinforced  by  400  ^ns.   10^-  ounces. 
men  ?         Jns.  6|  Months. 


132     SINGLE  RULE  of  THREE  INVERSE.    Sect,  II.  7. 

8    If  a  man  perform  a  purney   in  9.  If  a  piece  of  land,  40  rods  in 

I5days,when  the  day  is  12hours  long,  length,   and  4  in    breadth    make  an 

in  how  many  will  he  do  it  when  the  acre,  how   wide  must  it  be,  when  it 

day  is  but  10  hours  ?    ^«a.  18  days,  is  but  25  rods  l®ng  ?     jins.  6|  rods. 


10,  There  was  a  certain  building 
raised  in  8  months  by  120  workmen, 
but  the  same  being  demolished  it  is 
required  to  be  rebuilt  in  2  months  :  I 
demand  how  many  men  must  be  em- 
ployed about  it  ?       Jns.  480  men. 


\\.  How  much  in  length,  that  is 
3  inches  broad, .will  make  a  square 
foot?  jUns.  ASinches. 


1 2.  There  is  a  cistern, having  1  pipe  13.  If  a  field  will   feed  6    cows 

which  will  empty  it  in  10  hours;  how  91  days,   how  long   will   it  feed  21 

many  pipes  of  the  same  capacity  will  cows  I  Ans  26  days. 

empcy  it  in  24  minutes  ?  Ana. 2  5  pipes. 


Sect.  II.  7.    SINGLE  RULE  of  THREE  INVERSE.     133 

GENERAL  RULE 

For  stating  all  questions  ivhether  direct  or  inverse. 

1.  Place  that  number  for  the  third  term,  which  signifies  the  same  kind 
of  thing,  with  what  is  sought,  and  consider  whether  the  number  sought  will 
be  greater  or  less.  If  greater  place  the  least  of  the  other  terms  for  the  first  ; 
but  if  less,  place  the  greater  for  the  first,  and  the  remaining  one  for  the  se- 
cond term. 

2.  Multiply  the  second  and  third  terms  together,  divide  the  product  by 
the  first,  and  the  quotient  will  be  the  answer. 

EXAMPLES. 

1.  Jf  30  horses  plough  12  acres,  how  many  will  forty  plough  in  the  same 
time  ? 

OPERATION. 

H.      H.        Ac.  Here  because  the  thing  sought  is  a  number  of 

30  :  40  :  :   12  acres  we  place  12,  the  given  number  of  acres, 

12  for  the  third  term  ;  and  because  40  horses  will 

plough  more  than  12,  we  make  the  lesser  num- 

30)48O(  1 6  Ans.  ber,  30,the  first  term  and  the  greater  number,  40 

the  second  term. 

2.  If  40  horses  be  maintained  for  a  certain  sum  on  hay  at  5  cents  per  stone, 
how  many  will  be  maintained,  on  the  same  sum,  when  the  price  of  hay  rises 
to  8  cents  per  stone  ? 

C       C        H.  Here,  because  a  number  of  horses  is  sought, 

8  :   5  :  :  40  we  make  the  given  number  ot  horses,  40,  the 

40  third  terra,  and   because  fe^ver  will  be  main- 

tainedfor  the  same  money,  when  the  price  of 

8)200(25  Ansiver.    hay  is  dearer,  we   make  tlie  greater  price,  8 

16  cents,  the  first  term,  and  the  lesser  price,  ^ 

— —  cents  the  second. 
40 
40 

The  first  of  these  examples  is  direct,  the  second  inver&e. 

Every  question  consists  of  a  supposition  and  a  demand. 

In  the  first  the  supposition  is,  that  30  horses  fUongh  1 2  acres,  and  the  demand 
hoiv  many  40  willfilough  ?  and  the  first  term  of  the  proportion,  30,  is  found  in 
the  supposition,  in  this  and  every  otiier  direct  question. 

In  the  second,  the  supposition  is  that  40  horses  are  waintained  on  hay  at  5 
cents  per  stone,  and  the  demand,  hoiv  many  will  be  jnaintaincd  on  hay  at  8  Cktits  ? 
and  the  first  term  of  the  proportion,  8,  is  found  in  tlie  demand,  in  this  and 
•very  other  inverse  question. 

3.  If  a  quarter  of  wheat  afford  60  4.  If  in  12  months,  100  dollars  gain 

tcnpenny  loaves,  how  many  eight  pen-  6  dollars  interest,  what  will  gain  the 

ny  loavea  may  be  obtained  from  it  ?  same  sum  in  5  months  ? 

Ar.s.  75  luavcff.  Afi.sr.'rr,  MO  dollars. 


134  Supplement  to  the  SING.  R.  of  THREE.  Sect.  IL  7: 


I 


Supplement  to  the  ^illgtc  ^Hufe  Of  <^\jttL 

mm Y " 

QUESTIONS. 


1 


1.  WHAris  the  Single  Rule  of  Three  ;  or^  the  Rule  of  Proportion  ? 

2.  How  many  kinds  of  Profiortion  are  there  ? 

3.  WHAfis  it,  that  the  Single  Rule  of  Three  Direct  teaches  ? 

4.  How  can  it  be  knovfn,  that  a  question  belongs  to   the  Single  Rule  oj  Three 

Direct  ? 

5.  WHAfis  understood  by  more  requiring  more,  and  less  requiring  less  ? 

6.  JFTbware  questions  in  the  Rule  of  Three  stated  ? 

7.  Ha  vjng  stated  the  question,  how  is  the  answer  found  in  direct  Profiortion  ? 

8.  JVhat' da  you  observe  of  the  frst  and  third  terms  concerning   the  different 

denominations,  sometimes  contained  in  them  ,? 

9.  When  the  second  term  contains  dijfferent  denominations  %vhat  is  to  be  done  ? 
\0.  How  is  it  known  what  denomination  the  quotient  is  oJ  ? 

1 1.  If  the  quotient,  or  answer,  be  found  in  an  inferior  denomination,  what  is  t9 

be  done  ? 

12.  When  the  terms  are  given  in  Federal  Money,  how  is  the  ofieration  conducted? 

13.  How  are  sums  in  Federal  Money  reduced  to  the  same  denomination  ? 

14.  When  any  number  of  barrels,  bales,  or  pieces,  Ijfc.  are  given,  what  is  the 

method  of  procedure  ^ 

15.  WHAfis  it  that  the  Single  Rule  of  Three  Inverse  teaches  ? 

16.  How  are  questions  stated  in  Inverse  Proportion  ? 

17.  What  is  understood  by  ^ore  rej^uirinc  less  Isj"  less  req^uirjnq  more  ? 

18.  How  is  the  answer  found  in  the  Rule  of  Three  Inverse  ? 

1 9 .  Wha  f  is  the  general  Rule  for  stating  all  questions  whether  Direct  or  Inverse  2 

EXERCISES. 

1.  If  my  horse  and  saddle  arc  worth  18  guineas  and  my  horse  be  worth 
liix  times  as  much  as  my  sriddle,  pray,  wjiat  is  the  value  of  my  horse  ? 

Jnswer  7Z  dollars. 


Sect.  II.  7.  Supplement  to  the  SING.  R.  or  THREE.  155 

2.  How  many  yards  of  mattin,  that  3.  Suppose  800  soldiers  were  placed 
is  half  a  yard  wide,  will  cover  a  room  in  a  garrison,  and  their  provtsions 
Wiat  is  18  feet  wide,  and  30  feet  were  computed sufficent  for  2 months; 
long?  Ann.  \'iO  yards.  how  many  soldiers  must  depart,  that 

the   provisions    may  serve    them   5 
mouths  ?  Ana.  480. 


4.  I  borrowed  185  quarters  of  corn  when  the  price  was  19«.how  much^ 
most  I  repay,  to  indemnify  the  lender,  when  the  price  is  17*-  4rf. 

Am,  202|J. 


136  Supplement  to  the  SING.  R.  of  THREE.  Sect.  II.  7. 

6.  A  and  B  depart  from  the  same  place  and  travel  the  same  road  ;  but  A. 
goes  5  days  before  B  at  the  rate  of  20  miles  per  day  ;  B  follow  at  the  rate  of 
2i  miles  pei*  day  :  in  what  time  and  distance  will  he  overtake  A  ? 

J?is.  B  will  overtake  Am  20  days,  and  travel  500  miles. 

Here  two  statements 
will  be  necessary  ;  one  to 
ascertain  the  lime  and 
another  to  ascertain  th« 
distance. 


METHOD 

Of  assessing  towji  or  parish  taxes. 

1.  An  invcjitory  of  the  value  of  all  the  estates,  both  real  and  personal,  and 
the  nuniber  of  polls,  for  which  eacli  person  is  rateable,  must  be  taken  in  sepa- 
rate tuiumns.  Then  to  know  v\h::it  must  be  paid  on  the  dollar,  make  the 
total  value  of  the  inventory  the  first  term  ;  the  lax  to  be  assessed,  the  second  ; 
and  1  dollar,  the   third,  and  the  quotient  will  shew  the  value  on  the  dollar. 

JVora.  lids  method  is  taken  from  Mr.  Pikf.''s  Arithmetic^  "Milk  tins  differ-- 
encCf  that  here  the  money  is  reduced  to  Federal  Currency, 


SicT.  II.  7.  Supplement  to  the  SING.  R.  of  THREE.   137 

2.  Make  a  table,  by  muUiplying  the  value  on  the  dollar  by  1,2,  3,  4,  5,  Sec. 

S.  From  the  Inventory  take  the  real  and  personal  estates  of  each  man, 
and  find  thsm  separately,  in  the  table,  Avhich  will  shew  you  each  man's  pro- 
J>ortional  share  of  the  tax  for  real  and  personal  estates. 

If  any  part  of  the  tax  be  averajjed  on  the  polls,  before  stating  to  find  the 
value  on  the  dollar,  deduct  the  sum  of  the  average  tax  from  the  whole  sum  to 
be  assessed  ;  for  which  average  miake  a  separate  column  as  well  as  for  the 
real  and  personal   estates. 

EXA.MPLES. 

Suppose  the  General  Court  should  grant  a  tax  of  1 50,000  dollars,  of  which 
a  certain  town  is  to  pay  Dolls.  3250,72  and  of  which  the  polls  being  624  are 
to  pay  75  cents,  each  ; — the  town's  inventory  is  69568  dollars  ;  what  will  it 
be  on  the  dollar  ;  and  what  is  A's  tax  (as  by  the  inventory)  whose  estate  is 
as  follows,  viz.  real  856  dollars  ;  personal  103  dollars  ;  and  he  has  4  polls  ? 

Pol.     Ctfi.       Pol.    Dolls. 

1.  As  1  :  ,75  :  :  624  :  468  the  average  part  of  the  tax  to  be  deducted 
from  ^3250,72  and  there  will  remain  ^2782,72 

Dolls.      Dolls.  Cis.  Dolls.  Cts, 

2.  As  69568  :  2782,  72  :  :   1   :  4  on  the  dollar. 


TABLE. 

Dolls. 

Dolls,  cts. 

Dolls. 

Dolls. 

cts. 

Dolls. 

Dolls 

1    is 

4 

20  is 

80 

300    is 

8 

2  — 

8 

30    , 

1 

20 

300  — 

12 

S  — 

12 

40  — « 

1 

60 

400  — 

16 

4  — 

16 

50  — 

2 

00 

500  — 

20 

5   — 

20 

60  — 

2 

40 

600  -^ 

24 

6  — 

24 

70  — 

2 

80 

700  — 

28 

7  — 

28 

80  — 

3 

20 

800  ■— 

52 

8  — 

32 

90  — 

3 

60 

900  — 

36 

9  — 

56 

100  — 

4 

00 

1000  — 

40 

10  — 

40 

Now  lo  find  what  a's  rate  will  be. 


His  real  estate  being  856  dollars  I  find  by  the   Ta- 
ble that   800  dollars    is  g32  cts.  . 
that  50     —  —  2 

that     6     —  —  0  24 


Therefore  the  tax  for  his  t*eal  estate  is  34  24 
In  like  manner  I  'find   the  tax 
for  his  personal  estate  to  be 

Hi*  4  polls,  at  75  cents  each,  are  3 


4   12 


Real 

Dolts.  Cm. 


Personal. 
Dolls.  Cts. 


Polls. 
Dolls.  Cts. 


Total. 
Dol/s.  Cts. 


34     24     I       4      12       I 
S 


36 


133  DOUBLE  RULE  OF  THREE.  Sect.  IL  8. 

§  8,  «©ouMe0u!eof€!jm. 

WiM  •'.'  '-.r  -.,.-  ^  v.c  -vK-  -i.?  ■■^' 

The  Double  Rule  of  Three,  sometimes  called  Compound  Proportion", 
teaches,  by  having  five  numbers  given  to  find  a  sixth,  which,  if  the  proportion 
be  direct,  must  bear  the  same  proportion  to  the  fourth  and  fifth  as  the  thiid 
does  to  the  first  and  second.  But  if  the  proportion  be  inverse,  the  sixth  num- 
ber must  bear  the  same  proportion  to  the  fourth  and  filth,  as  the  first  does  to 
the  st-xond  and  third. 

RULE. 

1.  State  the  question,  by  placing  the  three  conditional  terms  in  such 
order,  that  that  number  which  is  the  cause  of  gain,  loss,  or  action,  may  pos- 
sess the  first  place  ;  that  vvliich  denotes  space  of  time,  or  distance  of  place, 
the  second  ;  and  that  which  is  the  gain,  loss, or  action,  the  third." 

2.  "  Place  the  other  two  terms,  which  move  the  question,  under  those  of 
the  same  namu." 

"  3.  Then,  if  tlie  blank  place,  or  term  sought,  fall  under  the  third  place, 
the  proportion  is  direct,  therefore,  muliiply  the  three  last  terms  together,  for 
a  dividend,  and  the  other  two  for  a  divisor  ;  then  the    quotient  will  be  the 


answer 


"  4.  But  if  the  blank  fall  under  the  first  or  second  place,  the  proportion  is 
inverse,  wherefore,  multiply  the  first,  second,  and  last  terms  together,  for 
a  dividend,  and  the  other  two,  for  a  divisor  ;  the  quotient  will  be  the  answer.** 

EXAMPLES. 

1.  If  100  dollars  gain  6  dollars,  in  J  2  ninths,  what  will  400  dollars  gain 
in  a  months  ? 

Statement  of  the  questio7t. 
■■     D.     JSL        D. 

100   :   12  :  :   6    Terms  in  the  ftupfiosition,  or  conditional  terms* 
400   :     8  Terms  which  77iove  the  quesdon. 

Of  the  three  conditional  terms,  it  is  evident,  that  100  dolfars  p«t  at  inter- 
est is  that  omt,  which  is  the  cause  of  gain  ;  consequently,  100  dollars  must 
be  the  first  term  ;  and  because,  12  months  is  the  space  of  time  in  which  the 
gain  is  made,  this  must  be  the  second  term  ;  and  6  dollars  which  is  the  gain, 
the  third  term.  The  other  two  terms  must  then  be  arranged  under  those  of 
tlje  same  name. 

Now  as  the  blank  falls  under  the  third  place,  therefore,  the  question  is  in 
direct  proportion,  and  the  answer  is  found  by  multiplying  the  three  last  terms 
together  for  a  dividend  and  the  two  first  for  a  divisor. 


Then,  12l00)192l0O( 


OPERATION. 

100 ':    12   :    : 

6 

400        8 

8 

0(3 

3200 

12 

6 

Dolls.   16  Ansnver. 


JOO  Div.  19200  Dividend. 


2,  If  100  dollars  gain  6  dollars  in   12  months,  in  what  time  will  400  dol" 
hrsgain  16  ? 


Sect.  II.  8, 


DOUBLE  RULE  of  THREE. 


139 


OPERATION. 
D.       M.  D. 

100  ;   12  ;  .  6       Here  the  blank  falling;  under  the  second 
400  16  term,  the  proportion  is  indirect. 

6  12      Therefore  raultiply  the  first,  second  and 

..  last  terms  together  for  a  dividend,  and  the 

2400  divis.   192  other  two  for  a  divisor. 
100 


19200  divided.  M, 

Then.  24)00)   192|00(  8  Jns. 
192 


3.   A  FARMER  sells   204    doHars 
worth  of  grain,  in  5  years,  when  it  is 
sold  at  60  cents  per  bushel  ;  what  is 
it  per  bushel  when  he  sells  1000  dol- 
lars worth,  in  18   years,  if  he  sell  the 
same  quantity  yearly  ? 
Cu.     Y.         D. 
60  :  5   :  :  204     cU.m. 
18  :  :  1000  :  ^^\^  Ans. 


4.  If  7  men  can  reap  84  acres  of 
wheat  in  12  days  ;  how  many  men 
can  reap  100  acres  in  5  days  I 


M. 

7  : 


D. 

12   : 
5  : 


J. 
84     M. 

100     20  .^ns. 


140     Supplement  TO  THE  DO.  R.  OF  THREE.    Sect. 11.3. 

Supplement  to  the  vDOUMe  MlXlt  Of  Cj^tCet 

■MB  -  -  -;   -  "I"   -  -'  -  -  IT       

QUESTIONS. 

1.  WHAfis  the  Double  Rule  of  Three  ;  or  Compound  Propori'ion  ? 

2.  How  are  (juestiona  to  be  stated  in  the  Double  Rule  of  Three  ? 

3.  How  is  it  knoivn^  after  the  statement  of  the   question^  whether  the  firofior-T 

tion  be  direct  or  inverse  ? 

4.  When  the  firofiortion  is  Direct  ^  how  is  the  answer  to  be  found  ? 

5.  When  the  proliortion  is  Inverse j  how  is  the  answer  to  bejound  ? 

EXERCISES. 

1.  If  6  men  build  a  wall  20  feet  long,  6  feet  hig-h,  and  4  feet  wide  in  16 
days,  in  what  time  will  24  men  build  one  200  feet  long,  8  feet  high  and^S 
thick? 

The  solid  contents 
in  each  piece  of  wall, 
according  to  the  giv- 
en dimensions,  must 
|)e  found  before  stat^ 
ing  tl>c  question. 


Sect.  II.  8.  Supplement  to  the  DO.  K.  of  THREE.     Ul 

2.  If  the  frr ight  of   \2C'wt.  ^qra,  6/6.    275  miles,  cost   §27,78:  1k>\v  fap 
piay  eoCw/.  Syr*,  be  shipped  for  §234,78  I  4ns,  480  wtV?^, 


S.  An  usurer  put  out  75  dollars,  at  I      4.  If  7  men  can  make  84  rods  ©f 
Interest ;  and  at  the  end  of  8  months  j  wall  in   6  days  ;  in   what  time  wili 
received  for  principal  and  interest,  79  I  IQ  men  make  150  rods  ? 
dollars  ;  I  demand  at  what  rate  per  |  jins.  5^5  days, 

f  ent  he  received  interest  ?  I 

Am.  ^/icr  cent. 


X42     Supplement  TO  THE  DO.  R.  OF  THREE.    Sect.  II.  8- 


5.  If  the  freight  of  9hhd.  of  sugar,  each  weighing  \2Civt.  20  leagues,  cost 
<C16  :  what  must  be  paid  for  the  freight  of  50  tierces  ditto,  each  weighing 
J^Cw^  100  leagues  ? 


I 

it 

1 


Sect.  II.  9. 


PRACTICE. 


143 


§  9.  Practice. 


"  Practice  is  a  contraction  of  ihe  Rule  of  Three  Direct,  when  the  first 
term  happens  to  be  an  unit,  or  one  ;  it  has  its  name  front  its  daily  use  among 
Merchants  and  Tradesmen,  beingc  an  easy  and  concise  method  of  working 
most  questions,  whicii  occur  in  trade  and  business." 

Proof.  By  the  Single  Rule  of  Three,  Compound  Mulliplication,  or  by 
varvirig  the  parts. 

Before  any  advances  are  made  in  this  rule,  the  Learner  must  commit 
to  memory,  the  following 

TABLES. 
AUiquot^  or  coen  parts  of  Money, 


Pis.  of  a  shil.  of  a;C. 


s.  and     £. 


—     i     —  -1- 


■f 

"2 

— 

■.\ 

T 

— 

^? 

"4 

~~ 

is 

I 

I 

lis 

1 

_8 

T60 

^ 

_ 

1 

T^ 

z;^s 

tV 

""" 

sio 

__ 

I 

2^ 

4¥7r 

I 

4:f 

960 

Sc?.  is  the  stlm  of  4rf.  and  \cl. 

7d.        6cf.  and  \d. 

8rf.    is  twice        4(/. 

9rf.  is  the  sum  of  6d.  and  2d. 

\0d.     6rf-  and  Ad. 


Pts.  of  a  pound. 

*.     d.  is   £. 

0  —  I  Practice  admits  of  a  great  va- 
8  —  ^  riety  of  cases,  the  multiplicity  of 
0  —  ±  which  serves  little  else,  than  that 
0  —  -\  of  confounding  the  mind  of  the 
4  —  ^-  Scholar  ;  a  different  method  will 
6  —  I  he  pursued  here  and  the  whole 
8  — ^>_  comprised,  in  a  few  cases,  such  as 

4  — y^  shall  be  useful  and  easy  for  th« 
3  — j_  Scholar  to  bear  in  his  memory. 

0  — 2^  The  small  number  of  examples 
0  — 2T  "nder  each  case  will  be  made  up 
8  — 3^'^  in  the  Supplement  ;  this  will  lead 

5  — 4?g-  the  Scholar  to  a   more  particular 
0  2\  — ^'g-  consideration  of  them. 


10 
6 
5 

4 

n 
o 

2 

L 
1 

1 

1 

0 
0 
0 


OPERATIONS. 


Pounds, SbilL  Pence,  Farthings. 


Dollars,  Cents,  Mills. 


When  the  price  of  the  given  quan-  |  RULE. 

tity  is  1/;.  1*.  l(i  per  pound,yard.  Sec.  I        Multiply   the   quantity    by   the 
then  will  the  quantity  itselfbe  the  an-  I  price  of  1  pound,  yard,  8cc.  the  pro- 
swer  at  the  supposed  price.  There-  |  duct  will  be  the  answer, 
fore, 


CASE      1, 

WifES  the  firlce  ofUjd.  lb.  l:fc.  con^ 
9iat.i  of  farthrngft  only;  If  it  be  one 
farthing,  take  a  fourth  of  the  quantity; 
if  a  half  penny,  Xxxkc  a  half  ;  if  three 
farthings  take  a  half  and  a  fourth  of 
the  quantity  and  add  them.  This 
gives  the  value  in  pence,  which  must 
be  reduced  to  pounds. 


144 


PRACTICE. 


Sect.  II.  0. 


Founds t  SbilL  Pence,  Farthings, 

EXAMPLES. 

K  What  will   362   yards  cost,  at 
Jrf  per  yard  ? 

OPERATION, 

2)362 
\'2)\%\  pence. 


\5s.    Id.  jins. 

Here  the  quantity  stands  for  the 
price  at  one  penny  per  yard,  but  as 
two  farth.lngs,  are  but  half  one  penny, 
therefore  dividing;  the  quantity  by  2 
gives  the  price  at  half  a  penny  per 
yard,  which  must  be  reducetl  to  shil- 
lings; 

2.  What  will  554^  yards  cost,  at 
\d,  per  yard  ? 

OPERATION. 
d.  q. 

4)354     2 


12)88      2 
laAds     2  Am. 
3.  What  will  203  yards  cost  at  3;/, 
per  yard  I  Ans.  16*.  5^-^. 


4.  What  will  816  yards  cost  at 
.q.  per  yard?  *       Am.  17«. 


t)tllars.  Cents,  Mills. 

1.  What  will  362    yards  cost  at  / 
mills  per  yard  ? 


OPERATION. 

3   6  2  quantitij. 
,0  0  7  iirice. 


g2,  5  3  4  AmiDtr. 

JYorE.  The  answers  in  the  differ- 
ent kinds  of  money  will  not  always 
compare,  because  in  the  reduction  of 
the  price,  i\  small  fraction  is  often 
lost  or  gained. 


2.  What  will  3 54 J  yards  cost,  at 
3  mills  per  yard  ? 

OPERATION. 


3  5  4,5 


quantity. 


,0  0  3  firice. 


^10  )6   3  5  Ansiver. 

3.  What  will  263  yards  cost,  at  i 
cent  per  yard.  Ans,  %  2,63, 


4.  What  will   816  yards  cost  at  S 
mills  per  yard  ?         Ans.  %  2,448, 


S£CT.  II.  .9. 


PRACTICE. 


145 


Pounds-^  ShiL  Pence^  Farthings, 


5.  What  will  97  yards  cost  at  3y. 
per  yard  ? 


Ans»  68.  04, 

4 


6.  What  will  126  yards  cost  at  Id. 

per  yard  ?  Ans.  5s.  3d. 


CASE    2. 

Wmen  the  firice  of  Mb.  I  yard^  Isfc. 
consists  of  pence,  or  oj  pence  and  far- 
things i  if  it  be  an  even  part  of  a  shil- 
ling, find  the  value  of  the  given  quan- 
tity at  \s  per  yard,  (ihe  quantity  it- 
self  expresses  the  price  at  \s.per  yard; 
if  there  are  quarters,  Is^c.  tyrite  for  ^ 
od.for  \  &d.for  i  9d.J  and  divide  by 
that  even  part,  wliich  the  price  is  of 
1  shilling.  If  tlie  price  be  not  an  al- 
iquot or  even  part  of  1  shilling,  it 
must  be  divided  into  two  or  more  al- 
iquot parts  ;  calculate  for  these  sep- 
arately, and  add  the  values  ;  the  an- 
swer will  be  obtained  in  shillings, 
■which  must  be  reduced  to  pounds. 

T 


Dollars,  Cents^  Mills, 

5.  What  will  97    yards  cost,  at  1 
cent  per  yard?  Jins.  ,97  cents. 


6.  What  will  126  yards  cost  at  7 
mills  per  yard  ?  Jns.  gO,882. 


146 


PRACTICE, 


Pounds,  Sbili, Pence,  Farihings 

EXAMPLES. 

1.  What  will  476  yards  cost,  at 
7ld  per  yard  ? 

OPERATION. 

fl. 

^^,\    I  f  I  ^'\^  Price  at  \s. /ler  i/ard. 
^'28  firzce  at  6d./ieryard. 
59  6d./iriceac7{d./ieryd. 


Irfi 


2(o)29|7  Gd./irice  at  7\d.  fier  yd. 
^  14   17*.  6d.  jinswer. 

PROOF. 

1.  By  the  Rule  of  three. 

Y.        £.    s.    d.        Y. 

As  476  ;  14  17  6  ::  I 

20 

297 
12 

476)3570(7^. 


238 
4 

)952(2-/. 
952 


%  By  Compound  Multiplication, 

£.     s.     d. 

7^  price  of  1  yard. 
10 


6     3  firice  10  yards, 
10 


2      6  price  ef  100  yards. 

4 


12    10  0  price, of  4.00  yards. 

2     3  9  price  of  70  yards. 

3  9  price  of^  yards. 

r.\i>   17  €>  price  of  4.7 6  yards. 


Sect.  II.  £), 

Dollars,  Cents,  Mills, 

7.  What  win  476  yards  cdme  to 
at  10  cents  4  mills  per  yard  ? 

OPERATION. 

476 
,104 


1904 
4760 

^49,504.^^5. 
PROOT 

cts^  fn.  D.    ct8.  m.  yds. 
,1  0  4)4   9   5  0  4(476 
4    1    6 


7  9  0 

7  2  8 

6  2  4 

6  2  4 


Sect.  II.  9. 


PRACTICE. 


U7 


Founds,  ShilL  Pence,  Farthings^ 

2.  What  will   17G  yards  cost,  at 
9~_d.  per  yard  ? 


ed. 

5d. 
id. 


OPERATION. 
S. 

■J  I  176  value  at  \s.  fier  yd. 

^  !  88  value  at  ed.fier  yd. 
-|of  44  value  at  2d,  per  yd. 
7  4a?.  value  at  \d.iier  yd. 

2I0>13I9  ^d.—at'^\d.  per  yd. 
^6    \9s.Ad.  Jins. 

PR90J?. 


3.  What  will  5681  yards  cost  at  7d. 
per  yard?        Ans.€\&  \\s.  s^d. 


Dollars,  Cents,  Mills. 

8.  What  will  176  yards  cost,  at  IS 
cents,  2  mills  per  yard  ? 

Jns,  g23,232. 


4.  What  will  568^  yards  cost  at  9 
cents,  7  mills  per  yard  ? 

-^^w.  g  55,12. 


148 


PRACTICE. 


Sect.  11.9.^ 


Pounds,  SbilL  Pence,  Farthings. 

4.  What  will  685|  yards  come  to, 
at  '2>\d.  per  yard  ? 

Jns,  £7  2s.  lO^rf, 


Dollars,  Cents,  Mills, 


n 


10.  What  \7ill  685  2;  yards  come  to, 
at  3  cents,  5  mills  per  yard  ? 

4ns,  3S24,OOU 


5,  What  will  649  *  yards  cost,  at 
lOd.  per  yard  ?      Ms.  £27  U.  Ojrf. 


11.  What  will  6491  yards  cost,  at 
13  cents,  9  mills  per  yard. 

Jins.  S9Q,245, 


6.  What  will   6831    yards  cost  at 
Slf/.  per  yard  ? 

Jins.  jC23   10*.  0|f/. 


12.  What  will  683-1  yards  cost,  at 


Sect.  II.  9. 


PRACTICE. 


149 


Pounds.ShilL  Bence^  Farthings, 

CASE    3. 

If  the  price  of  Mb.  1  yard^  ifc.  be 
shillings  and pence^  and  an  even  part  of 
1  f^.  Divide  the  value  of  the  given 
quantity  at  \{^  per  yard  by  that  ex'en 
part,  which  the  price  is  of  ^Cl.  The 
quotient  will  be  the  answer. 

EXAMPLES. 

l.What  will  7 19 J  yards  cost   at 
1*.  4.i/.  per  yard  ? 

OPERATION. 
^.  Si 

I  1/4  I  -I5.  I  719    \0  price  atU  per  yd. 


143    1 Q price  at  As. per  yd. 


Dollars,  Cents,  Mills. 


Jns.  47  19  4d.  at  \f4  per  yd. 
Herk  for  the  sake  of  ease  in  the 
operation,  because  5X3:=:  15,  there-? 
fore  I  divide  the  price  at  one  pound 
per  yd.  by  5,  and  that  quotient  by  3, 
which  gives  the  answer. 

2.  What  will  648  yards  co»t,  at 
1/8  per  yard  ?  jins.  /C54. 


13.  What  will  7 19 J  yards  cost,  at 
22  cents,  3  mills  per  yard  ? 

jins.  g  160,448. 


14.  What  will  ;^S  yards  cost,  at 
27  cents,  8  mills  per  yard  ? 

Am,  g  180 j  144. 


ISO 


PRACTICE. 


Sect.  II. 


f'ourtdsy  SbilL  Pence  ^  Farthings^ 

3,  Whativiil  687|  yards   cost,  at 
5».  per  yard?  Ans.fAY   \7s.  6d 


CASE.    4. 

Whek  thefiriceof  1  yard,  i^c.  is 
shillings  f  or  shillings  pence  b  farthings 
and  not  an  even  part  of  \C.  Multiply 
the  value  of  the  qnaniity  at  \s.  per 
yard  by  the  number  of  shillings  ;  for 
the  pence  and  farthings  take  parts,  as 
in  Case  2.  the  results  added  will  give 
the  answer,  which  must  be  reduced 
to  pounds. 

If  the  price  be  shillings  onlt/y  and  an 
even  number  ;  multiply  by  half  the 
price  or  even  number  of  shillings 
for  one  yard,  double  the  unit  fig- 
ure of  the  product  for  shillings,  the 
remaining  6gures  will  be  pounds. 

Note.  When  the  quantity  contains 
a  fraction,  work  for  the  integers,  and 
for  the  fraction  take  proportional 
parts  of  the  rate. 

EXAMPLES. 

l.What  will  167^  yards  cost  at 
17*.  6rf.  per  yard  ? 

OPERATIOBT. 

*- 
I  167 
17 

1169 
167 

2839  price  at  1 7s.  per  yd. 
83  6 — at  ed.per  yd, 
8  9  price  of  \  yd. 


I  ^d- 


210)293(1   Zd. 
Jn».  xn46   11«.  Zd. 


Dollars f  Cents,  Mills. 

15.  What  will  687^  yards  cost,  a|: 
83  cents,  3  mills  per  yard  ? 

Ans,  8^72,687. 


16.  What  will  U7f  yards  cost,  at 
g2j916?  ^n«.  488,43 


Sect.  II.  9. 


PRACTICE. 


1^1 


Pounds,  ShilLPetice,  Farthings, 

2.  What  will   5482  yards  cost,  at 
\2s.  4^f/per  yard  ? 

jins,  £  3391    19*.  9d. 


What  will  614  yards  cost,  at  16s. 
per  yard. 

OPERATION, 

614 

8  /laff  the  firice. 


4912   double    the  Jirst  figure 
^491   4s.  ^ns.  [for  shill. 

4.  What  will   176  yards  cost,  at 
i25.peryard?         Jns,£lQ5   I2s. 


5.  What  will  36  yards  cost,  at  7#, 
6^.  per  yard.  ^«*.  X;i3   10*. 


Dollars,  Cents,  Mills. 


17.  What  will  5482  yards  cost,  at 
S2,063  per  yard  ? 


18.  What  will  614  yards  cost,  at 
S  2,667  per  yard  I  Ans.  g  1637,538 


\'^.  What  will   176  yards  cost,  at 
2  dollars  per  yard  I        Akm,  gSoSl 


20    What  will  36  yanlscost,at 
S  1,25  per  yard?  ^««,S45, 


\ 


152 


fRACTJCE. 


SUCT.   II.    9. 


Founds  f  SbilL  Pctice  ^Farthings, 

CASE     5i 

When  the  price  of  1  yard^  1  Ih.  IsfC. 
is  fioUhds^  shillings^  and  pence  ;  Mul- 
tiply the  quantity  by  the  poimds  and 
if  the  shillings  and  pence  be  an  even 
part  of  a  pound,  divide  the  given 
quanliiy  by  that  even  part ^  and  add 
the  quotient  to  the  product  for  the 
answer  ;  but  if  they  are  not  an  even 
part  of  \{^.  take  parts  of  parts  and  add 
them  together.  Or,  you  may  reduce 
the  pounds  in  the  price  of  1  yard, 
8cc.  to  shilling's  and  proceed  as  in  the 
Case  before. 

EXAMPLES. 

1.  What  will  59  yards  cost  at 
iC6  78^6d.  per  yard  ? 


55.  is  ^  of  ^1 


2/6  is  i^  of  5.9. 


OPERATION. 

59  valhe  at  £\  per  yd. 
6 

354 — at  £6  per  yard, 
14    15.9.  at  5s. per  yd, 
7    7  ed.at2/eper  yd. 


Jins.  £076     2   &'at  £   7s,  9d. 

2.  What  will    16 3   yards   cost,  at 
£2   8s.  per  yard  ?  Jns.  /?39i   4s. 


Dollars,  Cents^  Mills. 


21.   W^hat  will   59  yards  cost  at 
g21,25  per  yard  ? 


OPERATION. 

D. 

C. 

21,25 

59 

191 

25 

1062 

5 

gl253   75  Jins. 
22.   What  will   163  yards  cost  at 
8  dollars  per  yard  ?        Ans,  g  13,04 


Sect.  II.  9. 


PRACTICE, 


Pounds  y  SLilL  Pence  ^Farthings, 

3.   What  will   76   yards   cost,  at 
aCS  2s.  Id.  per  yard  ? 

OPERATION. 
». 

$d.  is  \  of  ,U«    76  value  at  Is.  per  yd. 

,^  ^  "  fizzzshiUs.  in  £3  Sa. 

152  value  at  "is.fier  yd. 
456  — At  608.  tier  yd. 
\d.  is  -J-  of  Ct/.     38 — ar  ed.fier  yd. 

6  4c/. — at  Id.  /ler  yd. 

2|0)475/6  4d. 

jins.£237    16*.  4<sf. 


4.  What  is  the  valuje  of  84  yards, 
at  £2  U*.  per  yard  ? 

./in*.  /C226   16«. 


Dollars,  Cents,  Mills, 


153 


23.   What  will   76  yards  cost  at 
SlO,43  per  yard? 

Jim.  g792,68. 


I 


24.  What  is  the  value  of  84  yards 
at  9  dollars  per  yard  ?      jins,  $756. 


154  SUPPLEMENT  to  PRACTICE.         Sect.  II.  9. 

Supplement  to  ^lUttXW 
QUESTIONS. 

1.    IVHAris firactke? 
3.   Wnris  it  so  called  ? 

3.  When  the  price  of  I  T/a?*^/,  ^c.  itt  ftir things ^  koto  is  the   value  of  any  given. 

quafitity  Jound  at  the  same  rate  7 

4,  When  the  firice  consists  of  pence  andfarthingn^  and  is  an  even  fiart  of  1  s« 

hoTV  is  the  value  of  any  given  quantity  found  ? 
* ,  5.    When  the  firice  is  pence  and  farthings  and  not  an  even  part  of  Is.  ivhat  is 
the  method  of  procedure  ? 

6.  When  the  price  consists  oj  shillings,  pence  and  farthings,  how  is  the  value 

of  any  given  quantity  found  ? 

7.  When  the  price  contains  shillings  and  pence  and  is  an  even  part  of  £>\  hovi 

is  the  operation  to  he  conducted  ? 

8.  When. the  price  consists  of  shillings  only,  and  an  even  number,   what  is  the 

most  direct  way  to  find  the  value  oj  any  given  quantity  ? 

9.  When  the  quantity  contains  fractions,  as  ^,  |)  ^^,  Ufc.  how  are   they   to  be 

treated  ? 

10.  When  the  price  consists  of  pounds,  and  lower    denominations,  how  is  the 

value  of  any  given  quantity  found  ? 

11.  When  the  prices  are  given  in  Dollars,  Cents  and  Mills,  how    is  the  value 

tfany  given  quantity  J  ound  in  federal  money  ? 
12.    WHAf  is  the  method  of  proof  ? 
{%„  How  are  the  operations  in  Federal  Money  proved  ? 

EXERCISES  IJV  PRACTICE.       . 

In  the  followirig  exercises,  the  attention  of  the  scholar  must  be  excited  first 
to  consider  t6  which  ot"  the  preceeding  cases  each  question  is  to  be  referred.. 
Thai  heini;  ascertained,  he  will  proceed  in  il^  operation  according  to  the  in- 
struction there  given. 

1.  What  will  745-^  ysfrds  cost  at  1  \d.  per  yard  ?         Jins.  x;34  3*.  7\d. 

tJNDER  which  of  the 
'  proceeding  cases  docs 

this  question   properly 
*  belong  ? 

^  What  must  be  done, 

with  the  fraction  (^  of  a 
yard)  in  the  quantity  ? 


H 


^*  # 


#■ 


Sect.  II.  9.         SUPPLEMENT  to  PRACTICE.  155 

i    2.  What  will  964  yards  cQSt,at  Is.  9d,  per  yard  ?  ^na.  £^0  6s.  8d. 

OPERATION.  PROOF. 


f-»    ^ 


#      % 


3,  What  will  354^  yards  cost,  at 
^d.  per  yard  ?       ^ns,  7.9.  A,^d  , 


4.  What  will  316  yards  cost,  at 
\d.  per  yard?        Aris^  19*.  9rf. 


-■■4 


5.  What  will  567^  ywds  GOSt>  at 
IJflf.  per  yard  ? 

Jns.cz  10*.  \\\d. 


6.  What  will  913J  yardacosc|Ntt 
Crf.  per  yard  ?  'r^ 

ATt9.  £22   16#.  W. 


156 


SUPPLEMENT  to  PRACTICE.       Sect.  II.  §. 


7.  What  will  912|  yards  cost,  at 
9cl.  per  yurcl  i 

Ma.  £34,  4».  4^d. 


8.  What  will  76  yards  cost,  at 
2d.  per  yard  ? 

y(f««.     125.    8i/. 


r 


I* '  # 


^  « 


9.  What  will  845   yards  co&t,  at 
8*.  per  yard  ? 


If).  What  will  9 1  ya»ds  come  lo 
at  1 6«.  per  yard  ? 

Jns,  £72   IGs. 


11.  What  will  156^  ya^ds  eotoc 
to,  at  6«.  4:d.  per  yard. 

jins.  £4t9   lU,  St/. 


12.  What  will  96   yards  cost  at 
10s.  U«/.  per  yard  ? 

^«*.  jC;48    12«. 


m 


^^ 


' 


Sect.  II.  9.         SUPPLEMENT  to  PRACTICE. 


157 


13.  What  will  67 1  yards  cost,  at 
12&.  2d.  per  yard  ?  ^w*.  ^41  1*.  2d. 


15.  What  will  75  yards  cost,  at 
£3  3«.  4^/.  per  yard  ? 

Jns,  £2o7   10«, 


14.  What  will  843  yards  cost,  at 
€«.  8J,  per  yard  ?        Jns.  £2&i. 


&^ 


16.  What  will  59  yards  come 
to,  «it  jp6  7s.  6d.  per  yard  ? 

.4ns.  £376  2s.  6d. 


y 


I 


17.  What  willj9^-   yards  come 
I    to,  *t/C3  ««.  8d.  per  yard? 
•  Jns.  /;199  3«.  4d. 


18.  What  will  68  yards  cost,  at 
/C  4  6*.    per  yard  ? 

Jm.  292  S<. 


V 


158  SUPPLEMENT  to  PRACTICE.         Sect,  II.  9? 

N.  B.  The  following  questions  are  left   "without  any  answers,  Chat   the 
Sciiolar  may  operate  and  prove  each  question. 

19.  What  will  11,  yacds  of  flannel,  at  %s.  ^d.  per  yard,  come  to  ? 

*    OPERATION'.  «fc  PROOF. 


12.  What  will  IS  lb  of  cotton  cost,  at  S*.  4c?.  per  Ife 


2 } ,  Wk  AT  will  1  &S  yards  of  ribbon  come  to,  at  &d.  per  yard  I 


-^.^ 

f 

l'*1k 

•>• 

^^     -■ 

' 

1 

-f^ 

* 

*» 


% 


THE  ^ 


SCHOLAR'S  ARITHMETIC. 


►  -r  -;::-  v15-  >r  >^  -r  *>  -j'.:-  a* « 


SECTION  III. 


I 


Rules  occasionally  useful  to  men  in  particular  callin^.^  and  pur- 
suits of  UJe. 


§  1. 1'nUoluttan. 

Tnvolutjon,  or  the  raising  of  powers  is  the  multiplying  of  any  given  num- 
ber intoitself  continually,  a  certain  number  of  times.  The  quantities  m  this 
way  produced,  are  called  powers  of  the  given  number.     Thus, 

4x4=:  16  is  the  2d.  power,  or  square  of  4.  r-4'* 

4X4x4zi64  is  the  3d.  power,  or  cube  of  4,  zr4^ 

4X4X4X4::=256  is  the  4th.  power,  or  biquadrate  of  4^=4-* 
The  given  number,  (4)   is  called  the  first  power  ;  and  the  small  figure,' 
which  points  out  the  order  of  the  povyer,  is  called  the  Index  or  the  Exfioncnt. 

■\ 

—  g>>i<o^^ 

§  2.  (iHboIution. 

Evolution,  or  the  extraction  of  roots,  is  the  operation  by  which  wc  find 
any  root  of  any  given  number.  •        > 

The  root  is  a  number  whose  continual  n\ultiplication  into  itself  prodoces 
the  power,  and  is  denominated  the  square,  cube,  biquadrate,  or  2d,  3d,  4th, 
root,  &cc.  accordingly  as  it  is,  when  raised  to  the  2d,  3d,  4lk,  Sec.  power,  equal 
to  that  power  Thus,  4  is  the  square  foot  of  16,  bccatise  4X'1:=:16  4  also 
is  the  cube  root  of  64,  because  4x4X'^=:64  ;  and  3  is  the  square  root  of  9, 
and  12  is  the  square  root  of  144,  and  the  kubc  root  of  1728,  bccaud« 
12x]2xl2:::;i72y,  and  soon. 


160  EXTRACTION  of  the  SQUARg  ROOT.  Sect.  III.  3 


To  every  number  there  is  a  foot,  although  there  are  numbers,  the  precis* 
roots  of  which  can  never  be  ebtained.  But,  by  the  help  of  decimals,  we  can 
approximate  towards  those  roots,  to  any  necessary  degree  of  exactness.  Such 
roots  are  called  Surd  Roors^  in  distinction  from  those,  perfectly  accurate, 
which  are  called  Rational  Roots.  ^1 

The  square  root  is  denoted  by  this  character  ij  placed  before  the  power  ;f 
the  other  roots  by  the  same  character,  with  the  index  of  the  root  placed  over 
it.      Thus,  the  square  root  of  16  is  expressed  ^  16,  and  the  cube  root  of  27 

3 

is  y/  27,  kc. 

\y;hen  the  power  is  expressed  by  several  numbers  with  the  sii^n  -|-  or— i 
between  them,  a  line  is  drawn  from  the  top  of  the  sign  over  all  the  parts  of 

it;  tlius,  the  second  power  of  21 — 5  is  v'^l — 5,  and  the  3d.  power  of  5  6-{-8 

3 

is  ^56  +  8,  Sec. 

Ihe  second,  third,  fourth,  and  fiftli  powers  of  the  nine  digits  may  be  seen 
in  the  following 

TABLE. 


Hoots,     - 

or  1st.  Powers. 

* 
1 

2 

4 

8 

16 

Squares, 

or  2d.  Powers. 

Cubes,     - 

or  od.  Powers. 

Bi quad  rates 

or  4th.  Powers. 

• 

Sursolids,     |or  5th.  Powers. 

32 

3|       4 

5 

6 

7 
49 

8 

9 

9       16 

25 

36 

64 

8.| 

27J      64 

125 

216 

343 
2401 

512 

729  ! 

81    256 

625 

1296 

4096 

6561 

243]l024 

3125 

777^ 

16807 

32768 

59049 

1 1 


§  3.  (ffitrattion  of  tijt  Ji^uare  Boot. 

To  extrnrt  the  square  root  of  any  number,  i.s  to  find  anothertiumber  which 
mulriplifd  by,  or  into  itself,  will  produce  the  given  number;  and  after  the 
root  is  found,  such  a  multiplication  is  a  proof  of  the  work. 

RULE. 

1.  *'  Distinguish  the  given  number  into  periods  of  two-  fic^ures  each,  by 
pulling  a  point  over  the  place  of  units,  another  over  the  place  of  hundreds,  and 
so  on.  which  points  shew  the  number  of  figures  the  root  will  consist  of. 

2.  "  Find  the  greatest  square  number  in  the  first,  or  left  hand  period,  place 
the  root  of  it  at  the  right  hand  of  the  given  number  (after  ihe  manner  of  a 
quouenl  in  division)  for  the  first  figure  of  the  root,  and  the,  square  number, 
under  the  period,  and  subtract  it  therefrom,  and  to  the  remainder  bring  down 
the  next  period  for  a  dividend. 

3.  "  PLy*  cE  the  double  of  the  root,  already  found,  on  the  left  hand  of  the 
dividend  for  a  divisor. 

^  4.  "  Seek  how  often  the  divisor  is  contained  in  the  dividend  (excpt  the 
riglit  hand  figure}  and  place  the  answer  in  the  root  for  the  second  figure  of  it, 
and  likewise  on  the  right  hand  of  the  divisor  ;  multiply  the  divisor  with  the  fig- 
ure last  annexed  by  the  figure  last  placed  in  the  rc>ot,  and  subtract  the  product 
from  the  dividend  ;  To  the  remainder  join  tlie  next  period  for  a  new  dividend. 


Sect.  III.  3.  EXTRACTION  or  the  SQUARE  ROOT.  161 

.5.  ^'  Double  the  fiojures  already  found  in  the  root,  for  a  new  divisor^  (or 
bringdown  your  last  divisor  for  a  new  one,  doubling  the  rii;ht  hand  figure  o.f 
it)  and  from  these,  find  the  next  figure  in  the  root  as  lust  directed,  and  con- 
tinue the  operation  in  the  same  manner,  till  you  have  brought  down  all  ilie 
periods.  .  .         . 

"  Note  1.  If, 'when  the  given  power  is  pointed  off  as  the  power  ijequires, 
the  left  hand  period  should  be  deficient,  it  must  nevertheless  stand  as.thp  first 
period.  .  ' 

"  Note  2.  If  there  be  decimals  in  the  given  number,  it  must  be  pointe*! 
both  ways  from  the  place  of  units  :  If,  when  there  are  integers,  the  first  pe- 
riod in  the  decimals  be  deficient,  it  may  be  completed  by  annexing  so 
many  cyphers  as  the.  power  require*  :  And  the  root  must  be  made  to  con- 
sist of  so  many  whole  numbers  and  decimals  as  there  are  periods  belonging 
to  each  ;  and  when  the  periods  belonging  to  the  given  number  are  exhausted, 
the  operauon  may  be  continued  at  pleasure  by  annexing  cyphers.'* 

EXAMPLES.    - 

I.  What  is  the  square  root  of  729  ? 

OPERATION. 

720(27  the  root. 

4  The  given  number  being  dislinguished  into 

periods,  I  seek  ilie  greatest  square  number  ia 
the  left  hand  period  (7)  which  is  4,  of  which 
the  root  (2)  being  placed  to  the  right  hand  of 
the  given  number,  after  the  manner  of  a 
quotient,  and  the  square  number  (4)  subtract- 
ed from' the  period  (7)  to  the  remainder  (3) 
I  bring  down  the  next  period  (29)  making 
for  a  dividend,  329.  Then  the  double  of  the 
root  (4)  being  placed  to  the  left  hand  for  a 
divisor,  I  say  how  often  4  in  32  ?  (excepting  9 
///(?  right  handJiguTeJ  the  answer  is  7,  which  t 
place  in  the  root  for  the  second  figure  of  it,  and 
729  also  to  the  right   hand   of    the  divisor  ;  then 

multiplying  the  divisor  thus  increased  by  the  figure  (7)  last  obtained  in  the 
root,  I  place  the  product  underneath  the  dividend,  and  subtract  it  therefrom', 
and  the  work  is  done. 

DEMONSTRATION 

Of  the  reason  and  nature  of  the  ^various  steps  in  the  extraction  oj  the 

Sq^UARE  Root. 
The  superficial  content  of  any  thing,  that  is,  the  rnunaber  of  square  feet, 
yards,  or  inches.  Sec.  contained  in  the  siirlace  of  a  thing,  as  of  a  tabic  or  floor, 
a  picture,  a  field,  ecc  is  found  by  mulliplying  the  length  into  the  breadth.  If 
the  length  and  breadth  be  equal,  it  is  a  square,  then  the  measure  of  one  of  the 
sides  as  of  a  room,  is  the  root,  of  which  the  superficial  content  in  the  floor 
oF  that  room,  is  the  second  power.  So  that  having  the  superficial  contents  of 
thefloor  of  a  square  room,  if  we  extract  the  square  r»ot,  wc  shall  have  the 
length  of  one  side  of  that  room.  On  the  other  hand,  having  the  length  of 
one  bide  of  a  square  room,  if  we  multiply  that  number  into  itself,  tha\  in  to 
I'aise  it  to  the  second  power,  we  shall  then  have  the  superficial  contents  of  the 
floor  of  that  room. 

The  extraction  of  tlie  square  root,  therefore   has  this  opperation  on  num- 
"S,  to  arrange  the  nuJiibcr  oj'vjhich  ive  txtruct  the  root  into  a  ftqtKirc  form.   As 

w 


I 


16^  EXTRACTION  or  ihe  SQUARE  ROOT.  Sect.  III.  3: 

if  a  tnaTi  should  have  625  yards  of  cnrpetirig;,  1  yard  wide,  if  he  extract  the 
square  root  ol  that  number  (625)  he- will  then  have  the  len^nh  of  one  side 
of  a  square  room,  tiie  floor  of  wh-ch,  62 'i  yards,  will  be  just  sufficient  to  cover. 

to  proceed  then  to  the  demonstration. 

Example  2.  Strppos-rxG  a  n<an  has  625  yards  of  carpeting,  I  yard  wide, 
what  will  be  the  fength  of  one  side  of  a  square  room,  the  Soor  of  which  his 
carpeting^  will  cover  ? 

The  frrst  s!sp  is  to  point  off  the  number  into  periods  of  two  figures  each. 
This  determines  the  number  of  figures  of  which  the  root  will  consist,  and  is 
done  on  this  princrple,  t^at  the  firoduct  of  any  tivo  nmnbtrs  can  have  at  most 
but  90  -many  filaces  ofjigurea  an  there  are  fiiacea  in  both  the  factors^  and  at  leasty 
but  one  less^oj  nvhich  any  fierson  may  satisfy  hinuelf  at  /lieasure^ 

OPERATPON, 

t>25(30  TSE  nurn'bef  berng  pointed  off  as  the  rule 

4  directs,  we   fi-nd  we  have   two   periods-  ;  conse- 

'  quently;   the   root  will    consist   of  two  figures. 

225  The  greatest    Sqilare  number  m    the    left  hand 

periiud  (6)15  4,  of  which  two  is  the  r»ot  ;  there- 

"^^Tcr.  T.  fore,  2  is  the   first  figure  of  the  root,  and  as  it  is 

-  certain  we  have  one  figure  more  to  find  in  the 
root,  we  may  for  the  presenc  supply  the  place 
of  that  figure  by  a  cyphei',  (20)  then  20  will  ex- 
press the  just  valae  of  that  part  of  the  root  now 
obtained.  But  it  mu<it  be  remembered,  that  a 
I'oot  is  the  side  of  a  square  of  eqiial  sides.  Let 
Us  then  form  a  square,  A,  Fig  I.  each  side  of 
which  shall  be  supposed  20  yards.  Now  the 
side  a  d  of  this  square,  or  either  of  the  sides, 
shews  she  roet,  20,  which  we  have  obtained. 


To  proceed  then  by  the  itite,  '■^  place  the  square  number  underneath  the  pC' 
riody  fubtracty  and  to  the  remainder  bring  down  the  next  period.^*  Now  the 
Square  number  (4)  is  the  superfi-cial  content  of  the  square  A— made  evident 
thtis, — each  side  ot  the  square  A,  measures  20  yards,  which  number  multi- 
plied into  itself,  produces  400,  the  superficial  contents  of  the  square  A  ;  also 
the  square  number,  or  the  square  of  the  figure  2  already  found  in  the  reot, 
is  4,  which  placed  under  the  period  (6)  as  it  falls  in  the  place  of  hundreds, 
is  in  reality  400,  as  might  be  seen  also  by  filling  the  pkces  to  the  right  hand 
with  cyphers,  then  4  subtracted  from  6  Sc  to  the  remainder,(2)  the  next  period 
(25)  being  brought  down,  it  is  plain,  the  sum  225  has  been  diminished  by 
the  deduction  of  400,  a  number  equal  to  the  superficial  contents  of  the 
s-quare  A. 

Henck,  Fig.  I  exhibits  the  exact  progress  of  the  operation.  By  the  ope- 
ration, 400  yards  of  the  carpeting  have  been  disposed  of,  and  by  the  figure 
is  seen  the  disposition  made  of  them. 

Now  the  square  A,  is  to  be  enlarged  by  the  addition  of  the  225  yards  which 
remain,  and  this  addition  must  be  so  made  that  the  figure,  at  the  same  time, 
shall  continue  to  be  a  complete  and  perfect  square.  If  the  addidon  be  made 
to  one  .side  only,  the  figure  would  loose  its  square  form  ;  it  must  be  made  to 
tO)o  iidefi  ;  for  this  reason  the  role  directs,  ^^  place  the  double  of  the  root  al^ 
reudy  found  on  the  left  hand  of  the  dividend  for  a  divisor"  The  double  of  the 
Toot  is  just  equal  to  two  sides  b  c  and  c  d  oi  the  square,  A,  as  may  be  seen  by 
what  follow*, 


Sect.  III.  3.  EXTRACTION  oy  the  SQUARE  ROOT.  163 

Oi'EiiATiON  continued. 


625(25 
4 


45)225 
225 


The -double  of  tlue  root  is  4  which  j>lacecl  for 
■a  divisor  in  place  of  tens  (for  it  inust  he  re 
member ed^  that  the  next  Jijure  in  the  root  it  to  be 
placed  before  it  J  is  in  reality  40,  equal  Xa  the 
,§ides  b  c  (20)  and  c  d  (^0)  of  the  square  A. 


OQO 


'0 


A 

20 

400 


C. 


30 
5 

100 


20 


T    5     h 


The  square  A 


Fio.  n.  AiiAii^,  hy  the  mle,  "  Seek 

y  5  ^ow  often  the   divisor  is  contain^ 

~^-        ~  20  5      I  f^'  ^*«  ^^^   dividend  feafe/it    the 

Q  5  D  5     J5  right  hand  f^nire  J  and /dace  the 

^  '  I QQ  ~25~^i  ananver  in  the  root^fctr  ihe    sec^ 

ond  figure  of  its  and  dpi   Hie  right 
Jhand  9f  the  dix'jsQr'* 

^w  If  t'he  sides  b  c  and  c  d 
of  the  square  A,  Fig.  II.  is 
the  lenj^th  to  "Which  the  remain- 
ing 225  yards  are  to  be  added, 
and  the  divisor  ^4 /ew.^J  is  the 
&\\m  of  these  two  sides,  it  is 
then  evjdcm,  that  225  divided 
by  the  leagth  of  the  two  sides, 
'that  is  by  tlve  idivisor  (A  tens} 
will  give  .the  breadth  of  this 
new  addition  of  the  .225  yards^ 
to  the  sideA  6  c  aod.c^  of  the 
«quarc,  A. 

But  we  are  directed  to  "  except  the  rig%t 
/land  figure ^'^  and  also  to  ^'' place  the  quotient 
— — —  figure  on  the  right  hand  of  the  divisor  ;"  the 

froofQ35  ijd9         reason  of  which  is  that  t\he   additions,  C>y^ 
and  C  g  h  to  the    sides   he  and    cd  of  the 
square,    A,  do  not  leave  the  figure  a  com- 
plete square,  but  there  is  a  deficiency,  D,  at  the  corner.     Therefore,  in  di- 
.  viding,  the  right  hand  figure  is  excepted,  to  leave -something  of  the  dividend, 
for  this  ileliciency  ;  and  as  the  deficiency,  D,  is  limited  by  the  additions  C  e  f 
and  C  .5-  /i,  and  as  the  quotient  figure  (5)  is  the  width  of  these   additions,  con- 
sequently equal  to  one  side  of  Ihe  square,  D  ;  therefore,  the  quotient  figure 
(5)  placed  to  the  right  hand  of  the  divisor  {^tensj  and  multiplied  into  itself, 
gives  the  contents  of  the  square,  1),  and  the  4  fcw.^rrito  the  sum  of  the  sides, 
be  and  cd  of  the  addition  Cef  and  Cgh^   multiplied    by    the  quotient  figure, 
(5)  the  width  of  those  additions,  give    the   contents  C  r/and    C.^  A,  which 
together  sub  traded  from  the  dividend,  and  there   being  no  remainder,  shew 
that  the  225  yards  are  di<*posed  in  the  new  additions  Cf /",  C^A,  and  D,  and 
the  figure  is  seen  to  be  continued  a  complete   square. 

Consequently, ^f  II.  shews  the  dimensiors  of  u  square  rcom,  C5  yard^ 
on  a  side,  the  floor  of  which,  tJ25  yards  of  carpeting,  1  yard  wide  will  be  suffi- 
cient to  cover. 

The  proof  is  seen  by  adding  together  the  different  parts  of  the  figtirr. 

Such  are  the  principles,  on  which  ihc  operation  of  extracting  the  square 
root  is  grounded. 


—400  yds. 

—  100— 
P^/i— 100 — 
D       —  25— 


1^4  E  XTRACTION  of  the  SQUARE  ROOT.  Sect.  III.  3. 

3.  What  is  the  square  root  of  4.  What  is  the  square  root  of 

1O3426'50  ?  Jns.  3216  43264?  ^/i*.  ?   208. 


5.  What  is  the  square  root  of  964,5192360241  ?  ^/z^,- 51,05671 


Sect.  III.  3.  EXTRACTION  of  the  SQUARE  ROOT.  169 


6  Wh 

99800 


iHkT  is  the   square    root  of  7.  "What   is  the   square  rpot  of 

1  i"  ^72*.  999.  ^y:T^S.4,jQ9V"  jinn,    15^3 


Zi  Wlvatis  the  square  root  of  ipqq892198,40ai.?      .    yins.  SSlQT^l 


166  SUPPLEMENTto  t«e  SQUARE  ROOT.  Sect.  III.  3. 

Supplement  to  the  ^quatC  ClOOt 

..    II—.  -:>  ^  iit  jfc  -irr  iVe  •»>  — i|ii . 

QUESTIONS. 

1.  JVxjristo  he  understQod  by  a  root  ?  A  Jio^er  ?  The  second)  thifd^   and 

fourth  fioivers  P 

2.  WnAfia  the  IndeXy  or  Exfionent  ? 

5.   If- HAT  is  it  to  extract  the  Square  Root  ? 

4f.   Why  is  the  given  sum  peinted  off  into  fieriods  of  tv)0  Jigures  each  ? 

5.  Ik  the  operation^  having  found  the  Jirst  figure  in  the  rooty  tvhy  do  we  sub- 

tract the  square  number^  that  is  y  the  tqucire  of  that  Jigurey from  the  peri- 
od in  ivhich  it  ty«*  taken  ? 

6.  Why  do  "JDe  double  the  root  of  a  divisor  ? 

7.  Ivi  dividing  why  do  mxe  except  the  right  hand  figure  of  the  dividend  ? 

fjt.   Why  do  ive  place  tfje  quotient  fgure  in -the  root  and  also  to  the  right  hand  of 

the  divisor  ^ 
9.  Ip  there  he  decimals  in  the  given  numbery  hoxv  mu^t  it  be  pointed  ? 
JO.  Noiv  is  the  operation  of  extracting  t^e  Square  Root  proved  ? 

EXERCISES  IN  THE  SQVARJ^  ROOT. 

i.  A  Clergyman's  glebe  consists  of  three  fields;  the  first  contains  5 
Jcr.  2  r  \2  p.  the  second,  2  ac.  2  r  15  p.  the  third  1  ac,  I  r.  14  fi.  in  ex* 
change  for  which  the  heritors  agree  to  give  hini  a  square  field  equal  to  all 
tl>c  ihree.  Sought  the  side  of  the  ^uare  I  Ana.  39  poles. 


2.  A  OENERAL  nas  an  arm)'  of  4(596 
men  ;  how  many  must  he  place  in  rank 
ar^d  file  to  form  them  into  a  square  ? 
Jnsiver  64. 


Sect.  III.  3.  SUPPLEMENT  to  the  SQUARE  ROOT.  167 

3.  There  is  a  circle  whose  diameter  is  4  inches,  what  is  the  diameter  of  a 
circle  4  lime*  as  large  ?  -^«*.  8  inches.  ^ 

Note.  Square  the  given  diameter,  multipiy 
this  square  by  the  given  proportion,  and  ihc 
square  root  of  the  product  will  be  the  diameter 
requ'ired.     Do  the  same  iR  all  similar  cases. 

If  the  circi»;  of  the  required  diameter  were  toi 
be  less  than  the  circle  of  the  given  diameter,  by 
a  certain  proportion,  then  the  square  of  the  giv- 
en diameter  must  have  been  divided  by  that 
prop«rtioa^ 


4.  There  are  two  circular  ponds  in  a  gentleitian's  pleasure  gtound  :  the 
diameter  of  the  less  is  100  feet,  and  the  greater  is  three  times  as  large. 
What  is  iis  diameter.  ^mivcry  173,  2-^ 


5.  If  the. diameter  of  a  circle  be  12  inches,  what  will  be  the  diameter  of 
another  circle,  half  so  large  i  Jint.  8,  48 -frnc>if#. 


168  SUPPLEMENT  to  the  SQUARE  ROOT.  Sect.  111,3. 

6.  A  wall'  is  36  feet  high,  and  a  ditch  before  it  is  27  feet  wide  ;  what  is 
the  length  of  a  ladder,  that  will  reach  to  the  top  of  the  wall  from  the  oppo- 
site sides  of  the  ditch  ?  Answer  4:5  feet. 

Note.  A  figure  of 
three  sides,  like  that  form- 
ed by  the  wall,  the  ditch 
and  the  ladder  is  called  a 
right  angle  triangle^  of 
which,  the  square  of  the 
hypotenuse,  or  slanting 
side,  (the  ladder)  is  equal 
to  the  sum  of  the  squiAres 
of  the  two  other  sides,  that 
is,  the  heighth  of  the  wall 
and  the  width  of  the  ditch. 


7.  A  LINE  of  o&  yards  will  exactly  reach  from  the  top  of  a  Fort  to  the  op- 
posite bank  of  a  river,  known  to  be  24  yards  broad  ;  the  height  of  the  wall  is 
i-equired  ?  Ansnver  26^%o-\-yards. 


Sect.  III.  4.   EXTRACTION  of  the  CUBE  ROOT.     169 

8.  Glasgow  is  44  miles  west  from  Edinburgh  :  Pceble?^  is  exactly  soutli 
from  Edinburgh, and  49  miles  in  a  straight  liae  from  Glascow  ;  whatis  the 
distance  between  Edinburgh  and  Peebles  ?  AiiHioer,,  2\^o-\-miles. 


§  4,  (iHjetraction  of  t&c  Culje  0oot, 

To  extract  the  Cube  Root  of  any  number  is  to  find  another  number,  which 
multiplied  into  its  square  shall  produce  the  given  number. 

RULE. 

1.  "  Separate  the  given  number  into  periods  of  three  figures  each,  by- 
putting  a  point  over  the  unit  figure,  and  every  third  figure  beyond  the  place 
of  units. 

2.  "  Find  the  greatest  cube  in  the  left  hand  period,  and  put  its  root  in  the 
quotient. 

3.  "  Subtract  the  cube  thus  found,  from  the  said  period,  and  to  the  re- 
mainder bring  down  the  next  period,  and  call  this  the  dividend.       0 

4.  "  MuLTU'LY  the  square  of  the  quotient  by  300,  calling  it  the  triple 
square,  and  the  quotient  by  30,  calling  it  the  triple  quotient,  and  the  sum  of 
these  call  the  divisor, 

5.  "Seek  how  often  the  divisor  may  be  had  in  the  dividend,  and  place 
the  result  in  the  qualient. 

6.  "  Mui.Tin.Y  the  tripple  square  by  the  last  quotient  figure  and  write 
the  product  under  the  dividend  ;  muliiply  the  square  of  the  last  (juolicnt  fig- 
ure i)y  the  triple  quotient,  and  place  this  product  under  the  last  ;  under  ulU 
sot  the  cube  of  the  list  qiioiioiil  fK^ure,  and  c.ill  tijcir  sum  the  subtrahend. 

7.  "  SuimiAcr  the  subtrahend  from  the  dividend,  and  to  the  remainder 
bring  down  tiie  neKt  period  for  a  new  dividend,  with  which  proceed  as  be- 
fore, and  so  on  till  the  whole  be   rmii.hed. 

Note.  The  same  rule  must  be  observed  for  continuing  the  operalion> 
and  pointing  for  decimals,  as  in  the  square  root." 

X 


170    EXTRACTION  ox  the  CUBE  RQOT.    Sect.  III.  4. 


Wh^t  is  the  cube  root  of  373248  ? 

Oi'ERATIO:?. 


Divisor  ; 49 10)30248 


373248(72  the  root. 

343  7^'7y^^00i=:\^700,the  tri/Ue  square, 

7y.oQ  z=.     2 1 G  r//e  trijile  qxiQtiau, 


14910 //;c 
29400 

S40  14700x2—29400 

^  2X2X210=:     840 

2X2X2     =         8 


J0248 


r  30248  the  sukrahend. 

00000     DEMONSTHATION 
OJ  theRecffif)!).  and  Kature  of  Ike  the  various  stefis  in  the  ojieratioii  of  extracting  the 

CUBE  ROOT. 
Any  solid  body  \\z.\i\-\^  Ri:x:  equal  sides^  and  each  of  these  sides  an  exact  (square 
is  a  Cube,  and  the  measure  ia  len?:lh  of  one  of  its  sides  is  the  root  of  that 
cube.  For  if  ihe  measure  in  teet  of  any  one  side  of  such  a  body  be  multipli- 
ed three  time«  into  itself,  that  is,  raised  to  the  tiiiid  power,  the  product  wiU 
be  the  number  of  solid  feet  the  whole  body  contains. 

AxD  on  the  other  hand,  if  the  cube  root  of  any  number  of  feet  be  extracted 
this  root  will  be  the  length  of  one  §ide  of  a  cubic  body,  the  whole  contents  oi 
which  will  be  equal  to  such  a  number  of  feet. 

Slpposincv  a  man  has  13824  feet  of  timber,  in  distinct  and  separate  blocks 
ef  one  foot  each  ;  he  wishes  to  know  how  large  a  solid  body  they  will  make 
when  laid  together,  or  what  will  be  the  length  of  one  of  the  sides  of  that  cu- 
bic body  ? 

To  know  thi?,  all  that  is  necessary  is  to  extract  the  cube  root  of  f hat  num- 
ber, in  doing  whicii  I  propose  to  illustrate  the  operation. 

OPEllATlOK. 

In  this  number,   pointed  off  as  the  rule 
15824(30  directs,  there   are  two  periods,  of  course 

^  there  will  be  two  figures  in  the  root. 


The  {greatest  cube  in  the  right  hand  pe- 
riod, (13)  is  S,  of  which  2  is  the  root,  there- 
fore, 2  placed  in  tlie  quotient  is  the  first  fig- 
ure of  the  root,  and  as  it  is  certain  we  have 
one  figure  more  to  find  in  the  root,  we  may 
lor  the  present  supply  the  place  of  that  one 
figure  by  a  cypher  (20)  then  20  will  express 
I  the  true  value  ©f  that  part  of  the  root  now 

^;  obtained.     But  it  must  be  remembered,  that 

V  ./  %^^^  cube  root  is  the  length  of  one  of  the  sides 

I  {^  .  _       _._:^i^  of  the  cubic  body,  whose  length,  breadth,  and 

A         20         F  tliickness   are   equal.    Let  us  then  form  a 

2C/  cube.  Fig.  I.  each  side  of  which  shall  be  sup- 

posed   20    feet;  now  the  side  A.  B.  of  this 

400  cube,  or  either  of  the  sides,  shews  the  root, 

20  (20)  which  we  have  obtained. 


Sl'l-i 

FiG. 

r. 

c 

'       n 

k 

5000  feetzzzthe  Qolid  contents  of  the  CuBEt 


Bect.  III.  4.    EXTRzlCTION  o?  fnE  CUBE  ROOT.     171 

The  Rule  next  directs,  subtract  the  cuhe^  thus  founds  from  the  stiicl  period 
and  to  the  remainder  bring  doion  the  next  fieriod^  ijfc.  Now  this  cube  (8)  is  the 
solid  contents  o(  the  figure  wc  have  in  representation.  Made  evident  thus — 
Each  side  of  this  figure  is  20,  whith  being  raised  to  the  5d  ])ower,  iliat  is, 
the  Itngth,  breadth  and  thickness  being  multiplied  into  each  other,  irives  the 
solid  contents  of  that  figurerrSOOO  feet.  And  the  cube  of  the  root,  (2)  which 
we  have  obtained  is  8,  which  placed  under  the  period  from  which  it  was  tak- 
en as  it  falls  in  the  place  o^  tHonsands^  is  8000,  equal  to  the  solid  contents  of 
the  cube  A  B  C  D  E  f, which  being  subtracted  from  the  given  number  of  feet, 
leaves  5824  feet. 

Hence  J'^ig.  L  exhibits  the  eiiact  progress  of  the  operation.  By  the  opera- 
tion 8000  feet  of  the  timber  are  disposed  of,  and  the  figure  shews  tlie  disposi- 
tion made  of  them,  into  a  square  solid  pile  Which  measures  20  feet  on  every 
side. 

Now  this  figure  or  pile  is  to  be  enlarged  by  the  addition  of  the  5  824  feet, 
which  remains  ;  and  this  addition  must  be  so  made^  that  the  figure  or  pile, 
shall  continue  to  be  a  complete  cube;  that  is  have  the  measure  of  all  its  skies 
equal. 

To  do  thi^  the  addition  must  be  made  equally  to  the  three  different  squares, 
or  faces  a,  c  and  d. 

ThE  next  step,  in  the  operation  is,  to  find  a  divisor  ;  and  the  proper  di- 
visor will  be,  the  number  of  square  feet  contained  in  all  the  points  of  the 
figure,  to  which  the  addition  of  the  5  824  feet  ia  to  be  made. 

Hence  we  are  directed  "  multiply  tke  sgtiare  of  the  quotient  by  300,"  tlie 
object  of  which  is,  tt*  find  the  superficial  contents  of  the  three  faces  a,  c,  ^>,  to 
wl»ich  the  addition  is  now  to  be  made.  And  that  the  square  of  the  quotiuiit, 
multiplied  by  300  gives  the  superficial  contents  of  the  fscesjn,  c,  b^  h  evi- 
dent from  what  follows. 

Side  A  B:zi20^  2  quolieiit  f^nra 

Side  AF  =  20  t'      ,.     ^                                       2 
f>o/  the  facey  a.  

Superficial  content— z40o}  4  iiic  sc^vare  rf  2 


00 


The  triple  square  1  'lOOz^the  sufier-  The  triple  sqiiarr  rf  \  2Q0-=ihe  super- 
ficial  contents  oj  the  faces^  a,  c,  and  3,      fcial  contents  (fthefarca  «,  c,  andb. 

The  two  sides  A  B  and  A  F  of  the  Here  the  quotient  figure  2,  is  prop- 
face,  a,  multiplied  into  each  other,  crly,  fwo /f?.'^  for  there  is  another  fig- 
give  the  superficial  content  of  c,  and  ure  to  follow  it  in  the  root,  and  the 
as  the  faces,  a,  c,  and  6,  ar  i  all  equal,  square  of  2,  standing  as  units^  is  4,  but 
therefore,  the  content  of  the  face,  a  its  true  value  is  20  (r//r  ddc  A  B)  of 
multiplied  by  3,will  give  the  contents  which  the  square  is  400,  w^hercfore 
ofa,  c,  and^.  lose    two  cyphers,   and  ihero  two  c<f' 

phers  are  annexed  to  the  figure  2. — 
Hence  it  appears,  that  %ye  square  the  quotient,  with  a  view  to  find  ilie  supci- 
ficial  content  of  the  face,  or  square  a  ;  wc  multiply  the  square  of  the  quotient 
by  3,  to  find  the  superficial  contents  of  the  three  squares,  ff,  r,  Sc  6,  and  two 
cyphers  are  annexed  to  the  3,  because  in  the  square  of  the  quotient  trjo  cy- 
phcnt  were  lost,  the  (juotient  requiring  a  cypher  before  it  in  order  to  express 
its  true  value,  which  wouhl  throw  the  quotient  (2)  into  the  place  of  tcm^ 
whereas  now  it  stands  in  the  place  of  units. 

Now  when  the  additions  are  made  to  the  squares  rr,  r,  and  /•,  there  will  evi- 
dently be  a  deficiency,  along  tht-  whole  length  of  the  sides  of  ihr  squares  be- 
tween each  of  the  addiiions,  which  mnsi  !)c  supplied  before  the  fipure  can  be 
a  complete  cube.  These  deficiencies  will  be  3,  as  may  be  seen.  Fig,  It.  u.n.n. 

Therefore  it  is,  that  we  are  directwil,  "  muUi*Uy  the  quotient  by  30  callitig 
It  (hr  triple  qxiotinit, 

1  lie  triple  quotient  is  liic  sum  of  the  three  lines,  or  sides  against  wliicK 


172     EXTRACTION  of  the  CUBE  ROOT.    Sect.  III.  4.     i 

fire  the  deficiencies,  r?,  77,  n,  all  which  rrjeet  at  a  point,  nigh  the  centre  of  the      ' 
figure.     This  is  evident  Jrotn  what  ibllows.  j 


The   deficiencies  are  3  in   number, 
they    are  the    whole     len|tii   of    the 
sides  ;  the  length  of  each  side  is  20 
fuel,  iherelore  20 
3 


2  quotient) 
30 

Triple  quotient  60  equal  the  length 
of  o  sides )  ijfc. 


Triple  quotient  eOznto  the  length  oj  3 
aides  ivhere  are  deficiencies  to  be  filed. 


Here,  as  before,  the  quotient 
lacks  a  cypher  to  the  right  hand, 
to  exhibit  its  true  value  ;  the 
quotient,  itself  is  the  length  of  one  ol  the  sides,  where  are  the  deficiencies  ;  it 
it  muUipiicd  by  3,  because  there  are  3  deficiencies,  and  a  cypher  is  annexed 
to  the  3  because  it  has  been  omitted  in  the  quotient,  which  gives  the  same  pro- 
duct, as  if  the  true  value  of  the  quotient,  20,  had  been  multiplied  by  3  alone. 

,,^  ,         ?  1200  the  triple  square. 

We  now  have  ^      g^  ^^^  ^,,;^,^  ^J^^.^^^^ 

The  sum  of  which,  1260  is  the  divisor,  equal  the  number  of  square 
feet  contained,  in  all  the  points  of  the  figure  or  pile^to  which  the  addition  of 
the  5  824  feet  is  to  be  made. 


OPERATION  continued, 

13824(24 ///<?  root, 
8 


Divis.  1260)5824  the  dividend. 

4800 

960 

64 


5824  subtrahend. 


This  fi^'ure  in  the  root,  (4) 
shews  the  depth  of  the  addition, 
on  every  point  where  it  is  to  be 
made  to  the  pile  or  figure,  rep- 
represented,  Fig.  I. 


0000 


1200  trifile  square. 

4  last  qnotient  figure. 


Fig.  II.  exhibits  the  additions  made 
to  the  squares  a,  c,  6,  by  which  they  are 
covered  or  raised  by  a  depth  of  4  feet. 

The  next  step  in  the  operation  is  to 
find  a  subtrahend  which  subtrahend  is 
the  number  of  solid  feet  contained  in  all 
the  additions  to  the  cube,  by  the  last  fig- 
ure 4. 

Therefore,  the  rule  directs,  "  multi- 
ply the  triple  square  by  the  last  quotient 
figure.'" 

The  triple  square,  it  must  be  remem- 
bered, is  the  superficial  contents  of  the 
faces  a,  c,  and  b,  which  multiplied  by  4, 
the  depth  now  added  to  those  faces,  or 
squares,  gives  the  number  of  solid  feet 
contained  in  the  additions  by  the  last 
quotient  figure  4. 


AdOO  feet)  equal  the  addition  made  to  the 
Fig.  I.  a  depth  of  A  fat  on  each. 


squares^   or  faces^  a,   r,  6,  oJ 


Sect.  III.  4.     EXTRACTION  of  the  CUBE  ROOT.    173 


Fig.  III. 


T4n 
60  trt'/ite  quotient. 


Then,  "  muU^fihjthe  square  of  the  last  quo' 
iieni  figure  by  the  trifile  quotient  "  This  is 
to  fill  the  deficiencies,  «,  «,  w,  Fig.  II.  Now 
thestt  dcficieijcies  are  Unfiled  in  length, 
by  the  length  of  the  sides  (20)  and  the  tiiple 
quotient  is  the  sum  of  the  length  of  the 
deficiencies.  They  are  limited  in  width 
by  the  last  quotient  figure  (4)  the  square 
of  which  gives  the  area,  or  superficial  con- 
tents at  one  end,  which  multiplied  into  their 
length,  or  the  triple  quotient,  which  is  the 
same  thing,  gives  the  contents 
of  those   additions  4/24,    4/2,   4n, 


1 6  square  of  the  last  quotient  figure.  Fig.  III. 


360 
60 


960/^^^  disposed  in  the  deficiencies,  betnveen  the  additions  to  the  squares, 
a,  c,  b.  Fig.  III.  exhibits  these  deficiencies  supplied.,  4^4,  4w,  472,  and 
discovers  another  deficiency  ivhere  these  approach  together^  of  a  cor" 
ner  wanting  to  make  the  figure  a  complete  cube. 


to  ' 

O   I 


Fig.  IV. 

20  4 


} 


20     F4« 

4 
4 

16 
4 


Lastly,  ^  cube  the  last  quotient  figure.' 
This  is  done  to  fill  the  deficiency  Tig. 
III.  left  at  one  corner,  in  filling  up  the 
other  deficiencies,  w,  w,  n.  This  corner 
is  limited  by  thos2  deficiencies  on  every 
side,  which  were  4  feet  in  breadth,  con- 
sequently, the  square  of  4  will  be  the 
solid  content  of  the  corner,  which  in 
Fig.  IV.  e,  e,  e,  is  seen  filled. 


Now  the  sum  of  these  additions  make  the  siib- 
irahend,  which  subtract  from  the  dividend,  and 
the  work  is  done. 


64  feet  disposed  in  the  corner^  e,  e,  r,  where  the  additions  njn,  n,  ap- 
firoach  together. 


Figure  IV.  shews  the  pile  which  13824  solid  blocks  of  one  foot  each,  would 
make  when  laid  together.  The  root  (24)  shews  the  length  of  a  side  Yig.  /. 
shews  the  pile  which  would  be  formed  by  8000  of  those  blocks,  first  laid  to- 
gether ;  Yig.  II.  and  Vig.  III.  shews  the  changes  which  the  pile  passes  Ihro' 
in  the  addition  of  the  remaining  5824  blocks  or  feet. 

Proof.  By  adding  the  contents  of  the  ficst  figure,  and  the  additions  ex- 
hibited in  th^  other  figures  together. 


174    EXTRACTION  of  the  CUBE  ROOT.     Sect.  III.  4. 

Feet. 

8000   Contents  of  Fig,  J. 

4800  addition  to  the  faces  or  squares  a,  c,  and  5,  Fig,  II, 
960  addition  to  fill  the  deficiencies  n,  n^  «,  Fig.  III. 
64  addition  at  the  corner  e,  e,  <?,  Fig.  IV.  where  the  additions  'sohichfill 
the  deficiencies  tzj  n,  «,  aj\}iroach  together. 


13834  Number  of  blocks,  or  solid  feet,  all  wliieh  are  now    disposed   in 
Fig.  IV.  forming  a  pile,  or  solid  body  of  timber,  24  feet,  on  a  side. 

StJCH  is  the  demonstration  of  the  reason  and  nature  of  the  various  steps  in 
the  operAlion  of  extracting  the  cube  root.  Proper  views  of  the  figures,  and 
of  those  steps  in  the  operation  illustrated  by  them,  will  not  generally  be  ac- 
quired without  some  diligence  or  attention.  Scholars,  more  especially  will 
meet  with  difficulty.  For  their  assistance,  small  blocks  might  be  tormed  of 
xvood  in  imitation  of  the  Figures,  with  their  parts  in  different  pieces.  By 
the  help  of  these.  Masters,  in  most  instances,  would  be  able  to  lead  their  pu- 
pils into  right  conceptions  of  those  views,  which  are  here  given  of  the  nature 
cf  this  operation. 

3.  What  is  the  cube  root  of  21024576?  Jns'suer,276, 


Sect.  III.  4.     EXTRACTION  of  the  CUBE  ROOT.     175 

i-  WjiAT  is  the  cube  root  of  253395r99552  i  Amwer,  6328. 


176     EXTRACTION  of  the  CUBE  ROOT.    Sect.  HI.  4. 

5.  What  is  the  cube  root  of  84,604519  ?  jinswevy  4yo9. 


6.  What  is  the  cube  root  of  2  I  jfnsr^er,  1,25-^ 


Sect.  III.  4.  EXTRACTION  ot  the  CUBE  ROOT.       177 
SitPPLEMENT  to  the  CuftC^  moot 


QUESTIONS. 

1.    tVHAt  is  a  cube  ? 

2.  WHAfis  understood  by  the  cube  root  ? 

3.  What  is  it  to  extract  the  cube  root  ? 

4.  In  the  ofieration  having  found  the  first  figure  of  ike  root,  why  is  the  cube  of 

it  subtracted  from  the  fieriod  in  'othich  it  was  taken  ? 

5.  Wnris  the  square  oj  the  quotient  multifilied  by  300  .? 

6.  Wht  is  the  quotient  multiplied  by  30  ? 

7.  IVnr  do  we  add  the  triple  square  and  the  triple  quotient  together^  and  the 

sum  of  them  call  the  divisor  ? 
8   To  find  a  subtrahend,  why  do  We  multiply  the    trifile  square  by  the  last  giio. 

tient  figure  P  the  square  of  the  last  quotient  figure  by  the  triple  quotient  ? 

Why  do  we  cube  the  quotient  figure  ?  Why  do  these  sums  added,  make 

the  subtrahend  P 
9.  ffojfr  is  the  operation  proved  ? 

EXERCISES  IN  THE  CUBE  HOOT. 

1.  If  a  bullet  6  inches  diameter  weigh  32lb.  what  will  a  bullet  of  the  same 
metal  weigh,  whose  diameter  is  3  inches  ?  jimwer  4  lb. 

JVotE,  «  The  solid 
contents  of  similar  fig- 
ures are  in  proportion 
lo  each  other,  as  the 
cubes  of  their  similar 
»ides,  or  diameters." 


Y 


178    SUPPLEMENT  TO  the  CUBE  ROOT.      Sect.  III.  4. 

2.  What  is  the  side  of  a  cubical  mound  equal  to  one  288  feet  long,  21& 
broad,  and  4 8^  high  ?-  Jns.  I44f<ret. 


3.  Therp  is  a  cubical  vessel,  whose  side  is  2  feet ;  1  demand  the  side  of  a 
vessel,  which  shall  contain  three  times  as  much  ?  Jns.  2 feet  10  inc/ies  and 
•|  nearly. 

JVotE.  Cube  the  given 
side,  multiply  it  by  the 
given  proportion,  and  the 
cube  root  of  the  product 
will  be  the  side  sought. 


Sect.  III.  5.  FELLOWSHIP.  179 


§  5,  f  citoto.^fii}!. 


Fellowship  is  a  rule  by  which  merchants,  and  others,  trading  in  partner- 
ship, compute  their  particular  shares  of  the  gain  or  loss,  in  proportion  to 
their  stock  and  the  time  of  its  continuance  in  trade. 

It  is  of  two  kinds,  single  and  double. 

Singh  Fellowships 

Is  when  the  stocks  are  employed  equal  times, 

RULE. 
As  the  whole  sum  of  the  stocks  is  to  the  whole   gain  or  loss,  so  is  each 
man's  particular  stock  to  his  particular  share  of  the  gain  or  loss. 

Proof.  Add  all  the  shares  of  the  gain  or  loss  together  ;  and,  if  the  work 
be  right,  the  sum  will  be  equal  to  the  whole  gain  or  loss. 

EXAMPLES. 
1.  Two  merchants,   A  and  B,  make  a  joint  stock  of  200  dollars  :  A  puts 
in  75  dollars,  and  B  125  dollars^  they  trade  and  gain  50  dollars.     What  is 
each  man*s  share  of  the  gain  ? 

OPERATION. 

Dolls.  Dolls,  Doll^: 
jis  200  :  50  :  ;  75  As  200   :  50  :   :    125 

75  125 

250  250 

350  100 

' D.  CIS.  50 

200)3750(   18,75  A*s  share.  D  cfs: 

200  200)6250(31,25    B*a  share, 
600 


1750  

1600  250 

200 

1500  

1400  500 

400 

1000  \8,7B  A*s  share.  

1000  31^25  Ws  share.  1000 
1000 


50,00  /iroof, 

2.  Divide  the  number  560  into  4  such  parts,  which  shall  be  to  each  other 
as  3,  4,  5,  and  6. 

60"^ 
80'. 

100  rAus'-dJcr. 
120  J 

Q(hO  Proof. 


180  SINGLE  FELLOWSHIP.  Sect.  III.  5. 

3.  A  MAS  died  leaving  3  sons,  to  whom  he  bequeathed  his  estate  in  the 
following  manner,  viz.  to  the  eldest  he  gave  184  dollars,  to  the  second  155 
dollars,  and  to  the  third  96  dollars  ;  but  when  his  debts  were  paid,  there  wert. 
but  184  dollars  left  :  What  is  each  one's  propoation  of  his  estate  ? 

Ww5.  77,8291 

65,563  I s/igrcs. 
40,606} 


4.  A  and  B  compiuiicd  : — A  fiwt  in  £4,$.,  and  took  }  of  the  gain  ;  what 
<'}.:  B  t»ut  in  ?  Jns.  /J^O. 


Sect.  III.  5.  DOUBLE   FELLOWSHIP. 

Double  Fellowship. 


181 


Double  Fellowship,  or  Fellowship  with  time,  is  when  the  stocks  of  part* 
ners  are  coniinued  unequal  times. ' 

jRULE, 
Multiply  each  man -s  stock  by  the  lime  it  was  continued  in  trade.  Then, 
As  the  whole  sum  of  the  products  is  lo  the  whole  gain   or  loss,  so   is  each 
man's  particular  product  to  his  particular  share  of  the  loss  or  gain. 

|:XAMPLES. 
1.  A,  Ji  and  C,  entered  into  partnership:  A  put  in  85  dollars  for  8  months  ; 
B  put  in  60  dollars  for  10  months  ;  and  C  put  in  120  dollars  for  3  months  ; 
by  misfortue  they  lost  41  dollars  :  What  must  each  man  sustain  of  the  loss  ? 


85 
8 

080 


60 
10 

600 


OPERATION. 

120  680  A's  product. 

3  600  B's  product. 

—  360  C's  product. 


360 


4s   1640  :  41   ;   :  680 
'     680 

680 
2726 


64)0)2788(0(17  A's  loss, 
164 


1148 
1148 


1640 


Aa   1640  :  41 
600 


:  600 


164)0)246C10(15  B's  loss- 
164 


820 
«2Q 


0000 
As    1640  :  41  i  ;  360 
360 


2460 
123 


104jO)1476|O(9  C'sloss. 


1476 
0000 


Dolls. 
17  A*s  loss. 
15  B's  loss. 
9  C*s  loss. 

41   ProoL 


182  DOUBLE  FELLOWSHIP.  Sect.ULS. 

2.  A,  B,  and  C,  trade  together  :  A,  at  first  pot  in  480  dollars  for  8  months, 
then  put  in  200  dollars  more,  and  continued  the  whole  in  trade  8  months  lon- 
j!^er  ;  at  the  end  of  which  he  took  out  his  whole  stock  ;  B  put  in  800  dolls, 
for  9  months,  then  took  out  Dolls.  583,333  and  continued  the  rest  in  trade  3 
months,  C  put  in  ^566,666  for  10  months,  then  put  in  250  dollars  more, 
and  continued  the  whole  in  trade  6  months  longer.  At  the  end  of  their  part- 
nership, they  had  cleared  1000  doUaiij  j  what  is  each  man's  share  of  the  gain  ? 

4ns'(ver,     Dolls.  378,827  A's  share. 

320,452  B's  share. 

r--r-  300,721  C?  shave. 


Sect.  III.  5.    SUPPLEMENl'  to  FELLOWSHIP.  183 

.    Supplement  to  :fel[][oiX>0l)lp* 
QUESTIONS. 

1.  WRAf  is  Felloivshi/i .? 

2.  Of  Aow  mamj  kinds  is  Fellozvshifi  f 

3.  WuAr  is  single  Fellowshifi  ? 

4.  WHA'Tia  the  rule  for  operating  in  single  Felloiushifi  ? 

5.  Whav  is  double  Fellotuahifi  ? 

6;    WHAtis  the  rule  for  operating  in  double  Fellcwshi/i  ? 
7.  How  is  Felloiuship proved  ? 

EXERCISES  lA'  FELLOWSHIP. 

A,  B,  and  C,  hold  a  pasture  in  common  for  which  they  pzy£20fier  amrurrt^ 
In  this  pasture,  A  had  40  oxen  for  76  days  ;  B  had  36  oxen  for  50  days,  and 
C  had  50  oxen  for  90  days.  I  demand  what  part  each  ot  these  tenants  ought 
to  pay  for  the  jC20  ? 

£.  s.  d.  gr.  - 
Ans.  6   10  2   l||t§A*spart. 
3   J  7   I  Oi^no  B's  part. 
9   12  8  2|ffJ  CSparl, 


1(34  BARTER.  Sect.  III. 

§  6.  barter. 


iO-^o^o-^o  y  -.'.'f^o^o-^O'^o* 


Barter  is  the  exchanging  bf  one  toinmodity   for  another,  and  teaches 
merchants  so  to  proportion  their  quantities,  that  neither  shall  sustain  loss. 
Proof.     By  changing  the  order  of  the  question. 

RULE. 

1 .  When  the  quantiiy  of  ine  ccmmodity  is  ^iven^  with  its  valucy  or  the  value 
of  its  integer^  as  also  the  value  of  the  integer  of  some  other  commodity  to  be  ex- 
changed for  ity  to  find  the  quantity  of  this  commodity  :  Find  the  value  of  the 
commodity  of  which  the  quantity  is  given,  then  find  how  much  of  the  oth- 
er commodity  at  the  rate  proposed,  rhay  be  had  for  that  sum. 

2.  />  the  quantities  of  both  commodities  be  given,  and  it  should  be  required  to 
find  honv  much  of  some  other  commodity ,  or  honv  much  money  should  be  given,  for 

ihe  inequality  of  their  "valines  :  Find  the  sepjtrate  value  of  the  two  given  com-* 
inodities,  subtract  the  less  from  the  greater,  and  the  remainder  will  be  i\\b 
balance,  or  value  of  the  other  commodity.  < 

3.  If  one  com7nodity  is  rated  above  the  ready  money  fir  ice,  to  find  the  barter* 
i^S  firice  of  the  other  :  Say,  as  the  ready  money  price  of  the  one  is  to  the  bar* 
tering.price,  so  is  that  of  the  other  to  its  bartering  price. 

EXAMPLES. 

1.  How  milch  coffee,  at  25  cents  per  2.  I  have  760  gallons  of  mo- 
lb.  can    I  have  for  56  lb.  of  tea  at  43  lasses,  at  37  cents,  5  mills,  per 
cents  per  lb.  gallon,  which  I  would  exchange 
OPERATION.  for  66  Civt.  2qr.  of  cheese,  at  4 
5  6  lb.  of  tea.  dollars  fier  Civt.  Must  I  pay  of 
,4  3  /ler  lb.  receive  money  and  how  much  ? 

jins.  must  receive  19  dolls^ 


1    6  8 

2 

2   4 

^Ib. 

or. 

5)2 

4,0  8(96 

5^\  ans-iv 

2 

2   5 

1    5   8 

15  0 

8 

1   6 

2    5) 

1   2   8(5 
1    2   5 

Sect.  III.  6.  BARTER.  ISB 

3.  A  and  B»  barter  ;  A  has  150  bushels  of  wheat  at  5*.  9d.  per  bushel,  for 
which  B  gives  65  bushels  of  barley,  worth  2s,  lOd.  per  bushel,  and  the  bal- 
ance in  oats  *t  2s.  Jrf.  per  bushel  ;  what  quantity  of  oats  must  A  receive 
fr«m  B  ?  Amwer,  325^|  bushels. 


4.  A  HAS  liimen  cloth  worth  20rf.  an  Ell,  ready  itioney  ;  but  in  barter  he 
will  have  two  shillings ;  B  has  broadcloth  worth  14«.  6^.  per  yard  ready 
Taoney  j  at  what  price  ought  the  broad  cloth  to  be  rated  in  barter  i 

Amnvevy  17«.  4ci,  Sgr.  ^^  t^er  yard. 


186  SUPPLEMENT  to  BARTER.  Sect.  III.  6. 

Supplement  to  ^attCl% 
QUESTIONS. 

1.  WtiA<fis   Barter  ? 

2.  IVhen  and  how  does  this  rule  become  use/id  to  merchants  ? 

3.  When  a  given  quantity  of  one  commodity  is  bartered  for  some  other  com. 

modity^  hoiv  is  the  quantity  that  ivill  be  required  of  this  last  coTn?nodity 

found  ? 
A.  If  the  quantity  of  both  commodities  be  given  and  it  be  required  to  know  how 

much  of  some  other  coinmodity  ^  or  how  much  money  must  be  given  Jor  the 

inequality^  ivhat  is  the  method  vj  firocedure  ? 
5.  If  one  commodity  be  rated  above  the  motiey  firice,  how  do  you  proceed  to  find 

the  bartering  price  of  the  other  co7nmodity  7 
6.  How  is  Barter  proved  ? 

EXERCISES, 

\.  A  and  B  bartered  ;  A  had  41  Cwt.  of  hops,  SOs./ier  Cwt.  for  which  B 
gave  him  f.20  in  money,  and  the  rest  in  prunes  at  5d.per  lb.  I  demand  how 
many  prunes  B  gave  A  besides  the  ^20  \  Ana.  \7C.  3qi's.  4lb. 


2.  How  much  wine,  at  Sl.iB  per  guilon.  must  I  huve  for  26Cwt  2qr.  \4l6. 
of  raisins,  ?ii^9j4AA  per  Cwt. 

Ans.   196  gal.  \qt>  \pt.  and  \  very  nearly. 


Sect.  III.  7. 


LOSS  AND  GAIN. 


187 


§  7.  nm  anti  (Sain 


*<  Loss  and  Gain  is  a  rule  wliich  enables  merchants  to  estimate  their 
profit  or  loss,  in  buying  and  selling  goods  ;  also,  lo  raise  or  fall  the  pidce  of 
them,  so  as  lo  gain,  or  lose  so  much  per  cent/* 

CASE  I. 

7o  k?iow  what  is  gained  or  lost  fier  cent.  First,  find  what  the  gain  or  loss  i» 
by  subtraction  :  then,  as  the  price  it  cost  is  to  the  gainer  loss,  so  is  100 
■dollars  (or  C  l&O)  to  the  gain  or  loss,  per  cent. 

EXAMPLES. 

1.  If  I  buy  candles  at  16  cents,  7  2.  Bought  indigo,  at  g  1,20  per  lb. 

mills   per  lb.   and  sell  them  at  20  and   sold  the  same   at  90  cents    per 

cents  per  lb.   what  shall   I  gain  fier  lb.  what  was  lost/zer  cent  ? 

cent,  or  in  laying  out  100  dollars  ?  Answer,  25  dollar 9. 


OPERATION. 

I  sell  at  j20  per  lb. 
bought  at  ,167  per  lb. 

I  gain  ,033  per  lb. 


Then,  as,  167   :  ,0  3  3 
1  0  0 


100 


■D.  cis. 


,J67)3,  3  0  0(19,76  Ans. 
1   6  7 


16  3  0 
15  0   3 


12  7  0 
116  9 


0    1   0 
0  0  2 


3.  Bought  37  gallons  of  Brandy,  4.  Bought  hats  at  4«.  apiece,  and 

at  S  1>10  per  gallon,  and  sold  it  for  sold  them  again   at  4*9  ;  what  is  the 

840  :    what   was  gained  or  lost /<rr  profit  in  lajing  out  /;  100  ? 

i'cnl  ?         Arm.  ^\  J 19  less.  Ans. /i IS jiSt. 


188  LOSS  AND  GAIN.  Sect,  III.  7. 

CASE    2. 

To  knoiv  how  a  commodity  must  be  sold  to  gain  or  lose  so  much  per  cent.  As 
100  dollars  for  £\00j  is  to  the  priee  ;  so  is  100  dollars  for  ^100  J  with  the 
profit  added,  or  loss  subtracted,  to  the  gaining  or  losing  price. 

EXAMPLES. 

1.  If  I  buy  wheat  at  §51,25  per  bushel,  2.  If  a  barrel  of  rum  cost  15 

how  must  I  sell  it  to  gain  l5/ier  cent  ?  dollars,  how  must  it  be  sold 

OPERATION.  to  lose  \<d  ficr  cent  ? 

As  100  :   1,2  5  :  :    115  Ans,  gl3,50. 
1    I   5 

6  2  5 
12   5 
1  2   5 


■D.ets.m. 


100)1  4,  3  7  5(1,43  7  Jns. 
I  0  0 


4  3  7 
4  0  0 


3  7  5 
3  0  0 


7  5  0 
7  0  0 


5  0 

3.  If  120  lb.  of  steel  cost  £7  how  must  I  sell  it  per  lb  to  gain  iClSjper 
cent?  Jins»  IsAi  per  lb. 


Sect.  III.  7.  SUPPLEMENT  to  LOSS  and  GAIN.       189 

Supplement  to  %(i^^  anU  ^m. 

QUESTIONS. 

1.  What"  it  Loss  and  Gain. 

2.  Hjfjsg  the  iirice  at  which  goods  are  bought  and  sold^  hoiv  is  the  loss,  or 
gain  estimated  ? 

3.  To  know  how  much  a  commodity  must  be  valued  at  to  gain  or  lose  so  much 
fier  cent,  ivliat  is  the  method  of  procedure  ? 

4.  How  may  questions  in  Loss  and  Gain  be  proved? 

EXERCISES. 

\.  A  DRAPER  bought  100  yards  of  broadcloth  for/:56,     I  demand  how  he 

must  sell  it  per  yard,  to  gain  *C15  in  laying  out  jCIOO  ? 

Ans.   \2s.  \0d.  2q.  //, 


"2.  Bought  30  hogsheads  of  molasses,  at  600  dollars  ;  paid  in  dutiti 
*g20,66  ;  tor  freight  g40,7-8  ;  for  porterage  86,05  and  for  insurance,  S50,84  : 
If  I  sell  it  at  26  dollars  per  hogshead,  how  much  shall  I  gain  per  cent  ? 

^ns.  %\  1,695. 


190  DUODECIMALS.  Sect.  III.  8. 

§  8.  ©uotrctimal.^ ; 

OR, 

CROSS    MULTIPLICATION. 

mam   Tnr— nnr    ti — i    

This  rule  is  particularly  useful  to  Workmen   and  Artificers  in  casting  «p 

the  contents  of  their  work. 

Dimensions  are  taken  in   feet,  inches,  and  parts.     Iwches  and  parts  are 

sometimes  called  primes  ('),  seconds  ("))  thirds  ('")>  a^d  fourths  {""). 

TABLE.  By  this  rule  also  may  be  calculated  the 

12   Fourths  make  \    Third.  solid  contents  of  bodiesjharing  the meas- 

J2    lliirdfi     —      1    Second.  ures  of  their  different    sides,  and  is  very 

12  Seconds  —     1  /«c/t,or/*rfm<?,useful,  therefore,  in  measuring  wood. 
3  2  Inchesj  or  Pr.\   foot. 

RULE. 

1.  UvDEB.  the  multiplicand  write  the  correponding  denominations  of  the 
multiplier. 

2.  Multiply  each  term  in  the  multiplicand,  beginning  at  the  lowest,  by 
tile  feet  in  the  multiplier,  and  write  th*;  result  of  each  under  its  respective 
lerm,  observinp:,  to  carry  an  unit  for  every  12,  from  each  lower  denom- 
ination to  its  superior. 

3.  In  the  same  manner  multiply  the  multiplicand  by  the  inches  in  the  mul- 
tiplier, and  write  the  result  of  each  term  in  the  multiplicand  thus  multipli- 
ed, one  jilace  to  the  right  hand  in  the  product, 

4.  Proceed  in  the  same  manner  with  the  other  parts  in  the  multiplier, 
which  if  seconds,  write  the  result  two  filaces  to  the  right  hand  j  if  thirds,  three 
places.,  &c.  and  their  sum  will  be  the  answer  required. 

The  more  easily  to  comprehend  the  rule.  Note.  Feet  multifilied  by  Feet 

give  feet. — Feet  multifilied  by  Inches 
give  Inches.— ^Feet  multifilied  by  Sec^ 
onds give  Seconds. — Liches  multifilied 
EXAMPLES.  by  Inches  give  Seconds. -^Inches  mul- 

tifilied by  Seconds  give  Thirds. — .Sec* 
1.  MuLTirLY  7  feet,  3  inches,  2  Sec-  onds  m.ultijilied  by  Seconds  give  Fourths, 
ends,  by  I  foot,  7  inches,  and  5  Seconds. 


Here  I  multiply  the  7f.  Sin.  2"  by  the  1/.  in 
the  multiplier,  which  gives  seconds,  inches, 
and  feet. 

Next  I  multiply  the  same  7f.  Sin.  2'^  by  the 
7in.  saying  7  times  2  is  14  which  is  once  12 
and  2  over,  which  (2)  I  set  down  one  place  to 
the  right  i.and,  that  is  in  the  place  of  thirds,  and 
carry  1  to  the  next  place,  and  proceed  in  the  same 
I^rod.W     7     9    11      6  manner  with  ihe  other   terms.     Lastly  I   multi- 

ply the  multiplicand  by  the  3"  saying  3  times  2 
6  which  I  set  down  tvTO  places  to  the  right  hand  and  so   proceed  with  the 
other  terms  ot  the  multiplicand.  The  sum  of  all  the  products  is  the  answer. 


©PEl 

lATION. 

/''. 

I. 

/' 

7 

3 

2 

1 

7 

3 

7 

3 

2 

til 

4 

2 

10 

2   "" 

1 

9 

9      6 

r 


Sect.  III.  8.  DUODECLMALS.  191 


2.  3.  4. 

F.  L     p    J    „  F.  I.     p    J  F.  I.  J 


3     9 


27  9  9  Prod:       ^     ^  ?  25    6 /'/•of/.       ^     ^?52  3Pro(i 


5.  Multiply?/.  lz/2.9'^by  7/:  6.  Multiply  9/.  8m.  7"  by  12/ 

8m.   9"  oin.   10". 

Product  55/.  2m.  9''  3'"  9""-  Product  119/.  8'  2''  10'"  10"" 


7.  How  tniich  wood  in  a  load,  which  measures  10/.  in  length,  3/".  9m.  in 
%vidth,  and  4/  8m.  in  height  ;  and  how  much  will  it  cost,  at  I  dol.  33  cts.  per 
cord  ?  Jinsj  1  cord,  and  47  solid  feet  over  ;  it  ivill  coat  1  dQl,  81  cts.  8;/z. 


192  DUODECIMALS.  Sect.  III.  8. 

Or,  we  may  multiply  by  the  feet  as  already  directed,  and  for  the  inches, 
take  such  parts  of  the  multiplicand,  8cc.  as  the  inches  are  aliquot  or  even  parts 
of  a  foot,  as  done  in  the  rule  of  Practice. 

8.  How  many  square  feet  in  a  board  ot  16  feet,  4  inches  in  length,  and  2 
feet,  8  inches  wide  I 


Here,  in  the  first  place  I  multiply  the 
16/?.  4zn.   by  the   feet  (2)  of  the   multiplier; 
the  inches  (8)  not  being  an  even  part  of  a  foot, 
I  take  such  as  are  an  even  part  ;  thus,  6m.  is. 
''       half  a  foot,  therefore  divide  the  multiplicnnd 
8       by  2  for  6  inches,  and  that  quotient  by  3, (2m. 
is  \  0/6  inches)  for  2  inches,  all  which  being 
Jna,  43         6         8       added,  give  the  product  of  16   feet,  4  inches, 
multiplied  by  2  ft.  Sin, 

9.  Another  board  is  18  feet  9  inches  in  length,  and  2  feet,  6  inches  wide> 
how  many  square  feet  does  it  contain  ?  Jns.  46//.  10m.  6'\ 

B-y  J^raccice,  By  Duodecimals. 


OPERATION. 

Ft. 

in. 

6  inches 

18    \ 

16 

4 

2 

8 

32 

8 

% 

\ 

3 

2 

2 

8 

10.  Thcbe  is  a  stock  of  15  boards,  12  feet  8  inches  in  length,  and  13  inch« 
es  wide  j  how  many  feel  of  boards  does  thtc  stock  contain  ? 

Ans.  I^Sfeet^  \0  inches. 
By  Practice.  By  Duodecimals, 


Sect.  III.  8.     SUPPLEMENT  to  DUODECIMAtS.      159 

Supplement  to  ©UOtltCintaK,^* 

^mm  -Jl'r  •»;>  -:'.>  H^i  •:'.'?  4:v  iV-r  «■»- 

QUESTIONS. 

1 .  Of  'What  use  are  Duodecimals  ?  To  whom  more  esfiecially  are  they  usejnl  « 

2.  'In  what  are  dimensions  taken  ? 

3.  How  do  you  fiTOceed  in  the  multiplication  oj"  duodecimals  ? 

4.  For  what  number  do  you  carry  ? 

5.  WHAfdo  you  observe  in  regard  Xo  setting  down  the  fir  oduct  different  Jrorri 
what  is  common  in  the  multifilication  of  other  numbers  ? 

6.  Of  what  term  is  the  firoduct  which  arises  from  the   multifilication  of  Jfet  by 
inches  ?   Feet  by  seconds  ?  Inches  by  inches  ?   Inches  by  seconds  ?     Sec 
onds  by  seconds  ? 

7.  In  what  way  can  the  operation  be  varied  ? 

EXERCISES. 

1    Multiply  76  feet  3  inches  9  sec-         2.  What  is  the  product  of  371 
onds,  by  84  feet  7  inches  11  seconds.         feet  2  inches  6  seconds,  multiplied 

by  181/.  Un.  9", 

^n«.  67242/.  lOin.  V  4'"  6"" 


OPERATION. 

F. 

/.    " 

6  inches  is  ^)76 

3     9 

84 

7   11 

76X   4rr    304 

0     0 

76X    8—  608 

0     0 

3X84—       21 

0     0 

9X84—       .5 
I-H)               ^ 

3     0   '" 

1    10     6 

31)anrf  21)  3 

4     3      9' 

nft 

2      1    10 

6 

1 

7     0   11 

3 

1 

0     8      7 

6 

Vrod.  6460 

7      1      8 

3 

3.'  How  many  square  feet  in  a  stock 
of  12  boards,  17/  7'  long,  and  \f.  Sin, 
wide?  ^ns,  298/  U'. 


A  a 


many  cubic  feet  of  \vood 
6/    7'   long,   3/.   5'    high, 


4.  How 
in  a  load  ^ 
and  y.  8'  wide  ? 

Jns.  83/.  i'  »"  4'" 


1^4     SUPPLEMENT  to  DUODECIMALS.     Sect.  IIL  8. 

The  Dimensions  of  Waincoating,  Paving,  Plastering^,  and  Painting  arc 
taktn  in  Feet  and  Inches,  and  the  content  given  in  Yards. 

PAIN'2'ERS   AND    yoiNERS. 

To  find  the  Dimensions  of  their  work.,  take  a  line  and  apply  one  end  of  it  td 
any  corner  of  the  room,  then  measure  the  room  goint^  into  every  corner 
with  the  line,  till  you  come  to  the  place  where  you  first  began  ;  then  see 
how  many  feet  and  inches  the  string  contains  ;  this  call  the  Com/iass  or 
Rounds  which  multiplied  into  the  height  of  the  room,  and  the  Product  di- 
vided by  9,  the  Quotient  will  be  the  content  in  yards. 

EXAJtlPLES. 

1.  If  the  height  of  a  room  painted  2.  There  is  ^   room  wainscotted 

be    12/.  4m.  and  the   compass    84/.  the  compass  of  which   is  4 T/.  3' and 

1  \in.    How  many  square  yards  does  the  height  7f.  6'.  What  i^  the  content 

it  contain  I    Ansi  1 16  F.  3/  3'  8"  in  square  yards  ?  Ans.  39  Y,  3/  4'  6".. 


GLAIZERS    WORZ   BT   THE   ¥001', 


To  find  the  dimensions  of  their  ivorky  multiply  the  height  of  windows  by 
their  breadth. 


EXAMPLES. 


There  is  a  house  with  4  tiers  of  windows,  and  4  windows  in  a  tier  ;  the 
height  of  the  first  tier  is  6/  8'  ;  of  the  second,  5/,  9'  ;  of  the  third,  4/  6'  j 
and  of  the  fourth,  3/  10'  ;  and  the  breadth  of  each  is  3/.  5'  ;  What  will  the 
glazing  come  to,  at  19  cents  per  foQt  ?  Ans,  ^53,88. 


Sect.  III.  9.  ALLIGATION.  195 


§  9.  mum^m. 


wm^ ^ -r .;>  ^  -::5- ^ < 

Alligation  is  the  method  of  mixing  two  or  rr.ore  simples  of  different 
qualities,  so  that  the  composition  may  be  of  a  mean  or  middle  quality.  It  is 
•f  two  kinds,  Medial  and  Alternate. 

ALLIGATION  MEDIAL. 

Alligation  Medial  is  when  the  quantities  and  prices  of    several    things 
are  given,  to  find  the  mean  price  o»f  the  mixture  compounded  of  those  things. 

JRULE. 
As  the  sum  of  the  quantities  or  whole  composition  is  to  their  total  value, 
so  is   any  part  of  the  composition  to  its  valuie  or  mean  price. 

9 

EXAMPLES. 
1.  A  Farmeu  mingled  19  bushels  of  wheat  at  &s.  per  bushel,  and  40  bush- 
sels  of  Rye,  at  4s.  per  bushel,  and  12   bushels  of  barley,  at  3«,  per  bushel  to- 
gethe  r    I  demand  what  a  bushel  of  this  mixture  is  worth  ? 

©PERATiaN. 

Bush.                   s.       £•  s.  Bush.    £.     s.     Bush, 

19    Wheat,  at  6  is  5    14  As  71   :    15   10  ::   1 

40  Rye,       —4  —  8  20 

12    Barley,— 3-^1    16  


Sum  of  the  aimfiles  7 1  Total  value       15   10  284 

26 
2.  A  Refiner  having  5lb.  of  silver  bullion,  12 

ofSoz.  fine,  10/A.  of  7oz.  fine  and  15/6.  of  6oz.  

fine,  would  melt  all  together  ?  I  demand  what  )3 12(4</. 

i^neness  Mb.  of  this  mass  shall  be  ?  284 

Jns.6oz,  lo/iwt.  SgTs.j^ne*  — — 

28 
4 

3ll2(Iy. 
71 

41 


7\)3lQ(4s.4d.  l^q.Ms, 


196  ALLIGATION.  Sect.IIL9. 

ALLIGATION  ALTERNATE, 

Is  the  method  of  finding  what  quantity  of  any  number  of  simples,  whose 
rates  arc  given  will  compose  a  mixture  of  a  given  rate  ;  it  is,  therefore,  the 
rtvtrse  of  Alligation  Medial,  and  may  be  proved  by  it. 

RULE. 

1.  Write  the  prices  of  the  simples,  the  least  uppermost,  &c.  in  a  column 
under  each  other. 

2.  Connect  with  a  continued  liwe  the  price  of  each  simple  or  ingredient, 
which  is  less  than  that  of  the  compound,  with  one  or  any  number  of  those 
that  are  greater  than  the  compound,  and  each  greater  rate  or  price  M'ith  one 
or  any  jiumber  of  those  that  are  less. 

3.  Write  the  difference  between  the  mean  rate  or  price  and  that  of  each 
of  the  simples,  opposite  to  the  rates  with  which  they  are  connected. 

4.  Thkn  if  only  ohe  difference  stand  against  any  rate  it  will  be  the  quan- 
tity belonging  to  that  rate,  but  if  there  be  several,  their  sum  will  be  the  quan- 
tity. 

Note.  Questions  in  this  rule  admit  of  as  many  various  answers  as  there 
are  various  ways  of  connecting  the  rates  of  the  ingredients  together. 

EXAMPLES. 

A,GoLDSMiTii  would  iTiix  gold  of  1 8  carats  fine  Tvith  some  of  16,  19,  22 
and  24  carats  fine,  so  that  the  compound  may  be  20  carats  line  ;  what  quan* 
tiiy  of  each  must  he  take  ? 


f-16 ,7 

j   18 ,        2 

Mix  20  car.^  \  9-^    |       2 

j  22--'-^       2X1 
j^24 '  4   ' 


OPERATION.  PROOF. 

oz.  car.  fine. 

4ofgoldl6~]  16x4—64 

2  18  I  I8x2iz:36 

2  X^^Ana.       19X2~3S 

3  22'|  22x3=z66 

4 24j  24X4zz96 


oz. 


15   — 0,0  carats  fine    15)200(20  car.  fine. 


A  Druggist  had  several  sorts  of  Tea.  viz.  one  sort  at  12.9.  per  lb,  a- 
r  sort  at  1  Is.  a  third  at  95.  and  a  fourth  at  8s.  per  lb.  I  demand  how 
of  each  sort  he  must  mix  together,  that  the  whole  quantity  may  be  af- 


2, 
nother 
much  of  eacl 
forded  at  10*.  per  lb. 

lb.       s.fijb.  It.       s.fi.lb. 

r  3  at    12 

1  An8.    <:   "'    'I  2Ans.    )  ^  "'    '^  S  Jns. 

J  2  at     9 

Ls  at     8 

ib.       s.p.lb. 
fZ  at    12 

}  3  at     9 
(^2  at     8 

7  A>ht.  S'b.  of  each  sort. 

Note.   Thesf.  seven  x\vvswers  ariae  from  a 5  inany  different  Ways  of  Hn kin i 
the  liaiei  cf  the.  Simples  to:^'.'ther. 


SECT.IIL9.  ALLIGATION.  197 

CA.SE    2. 

When  the  raten  of  all  the  ingredients,  the  quantity  of  but  one  of  them,  and  the 
mean  rate  of  the  ivhole  mixture  are  gra  en  to  find  the  several  quantities  of  th:  r>:st, 
in  firafiortion  to  the  given  quantity  i  take  the  diftcrence  between  cm:  I),  r^ -ice 
and  the  mean  rate  as  before.     Then  say, 

As  the  difTerence  of  that  simple  whose  quantity  is  given, 

Is  to  the  given  quantity, 

So  is  the  rest  of  the  differences  severally  j 

To  the  several  quantities  required. 

EXAMPLES, 

1.  How  much  wine,  at  80  cents*  at  88,  and  92  cents  per  gallon  must  be 
mixed  with  four  gallons  of  wine  at  75  cents  per  gallon,  so  that  the  mixture 
may  be  worth  86  cents  per  gallon  ? 

OPERATION. 


!75 ^- ^         6+  2z=z  8  s(a?ids  against  the  given  quantity. 
80-,-f-^l         24-6z=8 
88-  + 
92--'- 


-\-J.-j\         6-f  11—17 
1+  6=17 

gal.       cts. 
4     at  80 


U7 


As8  :  4  :  :-(  17  :  s\ -^  ^8/ier.  gal.  The  answer, 
al  --  92 

2.  A  MAN  being  determined  to  mix  10  bushels  of  wheat  at  45.  per  bushel, 

with  rye  at  3s.  with  barley  at  2«.  afid  with  oats  at  Is.  per  bushel  y  I  demand 
how  much  rye,  barley,  and  oats  must  be  mixed  with  the  10  bushels  of  wheat, 
that  the  whole  may  be  sold  at  28c^.  per  bushel. 

B.  fi.                               CB.  CB. 

.       J    2  2  of  Bue       ^     .      J  40  0/  Bye         „  .       )    8  o/  Bye 

^     '^  0   of  Barley               \  S^  of  Barley  /  10  <2/  Barley 

2  of  Oats                   {_20  of  Oats  (  14  o/  Oats 

C  B.  11.  C  B. 

.      A      J   ^^  ^f  Rye           .J        3  12   2  of  Rye      >.  ^^„    ^    2  of  Rye 

<?/  Barley                    J    5  0  of  Barley  /  14,  of  Barley 

of  Oats                       K  17  2  of  Oats  ^  10  of  Oata  « 

f^Oo    Re 
'^-^""'Vrlfffallcy 
^20  of  Oa(s 


198  I         ALLIGATION.  Sect.IILS>. 

CASE    3. 

IVhen  the  rates  of  the  severalingredients,  the  quantity  to  be  comfiounded,  and 
tjte  mean  rate  of  the  nvhole  mixture  are  given  to  find  how  much  of  each  sort  ivili 
make  ufi  the  quantity  ;  find  the  differences  between  the  mean  rate,  Sec.  as  in 
case  1.  Then, 

As  the  sum  of  the  quantities,  or  differences, 

Is  to  tlie  given  quantity  or  whole  composition  ; 

So  is  the  difference  of  each  rate, 

To  the  required  quantity  of  each  rate. 

EXAMPLES. 

\.  How  many  gallons  of  water,  of  no  value,  must  be  mixed  with  brandy, 
a.t  one  dolUr  twenty  cents  per  gallon  so  as  to  fill  a  vessel  of  75  gallons,  that 
may  be  afforded  at  92   cents  per  gallon  ? 

OPERATION. 

Gal.  Gal.     Gal. 

5       0-^     28  GaL     Gal.         C  28  :    \7^qf  Water. 

^^^1,20--'     92         As   120  :  75  :  :   ^92  :  57^  cf  Brandy. 

Sum  120  75  given  quantity. 

2.  Suppose  I  have  4  sorts  of  currants  of  8^.  \2d.  \Bd.  and  32rf.  per  lb.  of 
v;liich  1  would  mix  1201b.  and  so  much  of  each  sort  as  to  sell  them  at  \6d, 
per  lb.  how  much  of  each  mu»t  I  take  ? 

f/d.     ai 
I  36 


-.1 


Jns.-{  12      ~     12  )>fierlb, 
I  24     -—     18  I 
,L48     —     22J 


o.  A  GitocER  has  currants  of  4rf.  6c?.  9d.  and  1  \d.  per  lb.  and  he  would 
make  a  mixture  of  240lb.  so  that  it  might  be  afforded  at  8rf.  per  lb.  how 
much  of  each  sort  must  he  take  ? 

f/6.  at     d.-^ 
|72-     4| 

jins.-^  24  —     6  )>fier  lb. 
48-     9\ 
*  ^96  —   llj 


Sect.  III.  ^.         SUPPLEMENT  to  ALLIGATION.      199 

Supplement  to  Alligations 


QUESTIONS. 

1.    WHAf  is  Alligation? 

S.  Of  haw  many  Idnda  is  jilligation  ? 

3.  WHAf  is  Alligation  Medial  ? 

4.  JVHAris  the  rule/or  o/ierating  ? 

5.  What" is  Alligation  ALfER^AtE  ^ 

6.  JVhes  a  number  of  ingredients  of  different  firiees  are  mixed  together^  hotO' 
do  we  firoceed  to  find  the  mean  firiee  of  the  comfiound  or  mixture  ? 

7.  Wheii  one  of  the  ingredients  is  limited  to  a  certain   quantity •,    tvhat  is  tfic 
method  of  procedure  ? 

8.  Wheh  the  whole  comfiosition  is  limited  to  a  certain  quantity,  how  do  you 

proceed  ? 

9.  How  is  Alligation  fir  oved  ? 


EXERCISES  i 

I.  A   Grocer  would  mix  three  \      2.  A  Goldsmith  has  several  sorts 

of  ijold  ;  some  of  24  caiatsfine,  some 
of  22,  and  some  of  18  carats  fine,  and 
he  would  have  compounded  of  these 
sorts  the  qtianiity  of  60  oz.  of  20  ca- 
rats fine  ;  I  demand  how  niach  of 
e^ch  sort  he  must  have  ? 
Ans,  12oz.  24  carats  fine,  \2  at  22  ca* 
rats  fine,  and  S6  at  18  carats  fine. 


sorts  of  sugar  together  ;  one  sort  at 
1  Orf.  per  lb.  another  at  7d.  and  another 
at  6d.  how  much  of  each  sort  must  he 
take  that  the  mixture  may  be  sold 
for  8f/.  per  lb  ? 
Ans.  311).  at  lOd,    2  at  Id.  and  2  at  dd. 


200  POSITION.  Sect.  III.  10. 

§  10.  po.^tion. 

Position  h  a  rule  whiclj,  by  false  or  supposed  numbers,  taken  at  pleasure^ 
<iiscovers  the  true  one  required.  '  It  is  of  two  kinds,  Single  and  Doubxe. 

Single  Position^ 

Is  the  working  with  one  supposed  number,  as  if  it  were  the  true  one,  to 
find  the  true  number. 

nULE. 

1.  Take  any  number  and  perform  the  same  operations  with  it  as  are  dcs- 
aribed  to  be  performed  in  the  question. 

2.  Then  say;  as  t!ie  sum  oftlvj  errors  is  to  the  given  sum,  so  is  the 
supposed  number  to  the  true  one  required. 

Proof.  Add  the  several  parts  of  the  sum  together,  and  if  it  agree  with 
the  sum,  it  is  right. 

EXAMPLES. 

1.  Two  men,  A  and  B,  having  found  a  bag  of  money,  disputed  who  should 
have  it  ;  A  said  the  half  third,  and  one  fourth  of  ihe  money  made  130  dollars, 
and  if  B  could  tell  how  much  was  in  it,  he  should  have  it  ail,  otherwise  he 
should  have  nothing  ;   1  demand  how  much  was  in  the  bag  ? 


OrERATlON. 

Sutifiose  60  dollars. 

'       As  65   :    150  :  :   60 

CO 

The  half    30 

65)7800(120  dollcrsi  the  ansiver. 

—  third     20 

65 

^fourth   15 

• 

130 

65 

130 

2.  A  B  and  C  talking  of  their  ages, 
B  said  his  atj,e  was  once  and  a  half  the 
age  of  A  ;  C  said  his  s^e  was  iwice- 
and  one  tenth  the  age  of  both  and 
that  the  sum  of  their  ages  was  93  ; 
"What  was  the  age  of  each  ? 
Ans.  yfs  I'ijB's  IS,  C".s  63  years. 


000 


A  PERSON  having  spent  J  and  | 


of  his  money,   had 
had  be  at  first  ? 


£26-  left  ;   what 
J/is  £160, 


4.  Seven  eights  of  a  certain  num- 
ber exceeds  four  fifths,  by  6  ;  what  is 
that  number?  Jns,QO. 


Sect.  III.  10.         DOUBLE  POSITION.  201 

Double  Position. 

Double  Position  is  that  which  discovers  the  true  number,  or  number 
sought,  by  making  use  of  two  supposed  numbers. 

BULE. 

1.  Take  only  tvvo  numbers  and  proceed  with  them  according  to  the 
condilions  of  the  question. 

2.  Place  each  error  against  its  respective  position  or  supposed  number  ; 
if  the  errorbe  too  great,  mark  it  with-f-  ;  if  too  small  with  — 

3.  Multiply  them  cross-wise,  the  first  pobition  by  the  last  error,  and  the 
last  position  by  the  first  error. 

4.  If  they  be  alike,  that  is,  hfixh  greater  or  both  less  than  the  given  num- 
ber, divide  the  difference  of  the  products  by  the  difference  of  the  errors,  and 
the  quotient  wHl  be  the  answer  ;  but  if  the  errors  be  unlike,  divide  the  sum 
of  the  products  by  the  sum  of  th«  errors,  and  the  quotient  will  be  the  answer. 

EXAMPLES. 

I.  A  MAN  lying  at  the  point  of  death,  le^t  to  his  three  sons  all  his  estate, 
viz.  to  F  halt  wanting  50  dollars  ;  to  G  one  ^)wrd  ;  and  to  H  the  rest,  wiiich 
was  10  dollars  less  than  the  sharu  of  G,  I  dCOT^nd  the  sum  Icfi,  and  each 
Son's  share. 

OPERATION. 

Supfiose  the  sum  300  dollars.         Again,  Sii/ifiose  the  snm  900  dotlars. 


Then,  3004-2—50—100  F's/iart.  Then,  900-^-2— 50— 400  F's  part. 

300-r  3  —  100  tr's/iart.  900-f-   3z=300  G's  part. 

G's  part  100-1 10=  90  H'spart  G'spart  200  —  1 0:=z29 0  Jrs  part. 

Sum  of  all  their  parts  390  Sum  of  all  their  parts  990 

Error  10 —  Error  904- 

Siipfiosi.         Errors.       Having  proceeded  with  the  sup- 

300^^-10 —  posed    numbers   according    to  the 

^C  conditions  of  the  question,   the  sujn 

900"^ ^90 4-  0/  all  their  parts  must  be  subtracted 

-^ from  the   supposed  number  ;  th'us, 

9000  27000  the  290  is  subtracted   from  300,  the 

27000  supposed  number.  Sec. 


.Dolls. 


Sum  of  the-}       iooyo(>000{Z\0  Answer, 
errors.  3 

The  divisor  is  the  sum  of  the  errors  90-|-and  10 — 

2.  There  is  a  fish  whose  Itend  is  10  feet  long  ;  his  tail  as  long  as  his  head 
and  hall'  the  length  of  his  bodvi  and  his  body  as  long  as  his  head  and  tail  ; 
what  if  the  whole  length  of  the  fish  ?  Ans.  UO/rrt. 


B  b 


202  DOUBLE  POSITION.  Sect.  III.  10, 

3.  A  CERTAiK  man  having  driven  his  Swine  to  raarkef,  viz.  Hogs,  Sovvs^ 
and  Pigs,  received  for  them  all  50/.  being  paid  for  every  liog  I8s.  for  every 
sow  \6s.  for  every  pig  2s.  there  were  as  many  hogs  as  sows,  and  for  every 
sow  there  were  three  pigs  ;  1  demand  how  many  there  were  of  each  sort  ? 

Jns,  35  fyogSj  25  sovfSi  and  75  pigs. 


A.  A  and  B  laid  oiit  equal  sums  of  money  in  trade  ;  A  gained  a  sum  equal 
to  -|-  of  his'stock,  and  B  lost  125  dollars,  then  A's  money  was  double  that  of 
B's  ;  What  did  each  one  lay  out  ?  Jns.  600  dollars. 


5.  A  and  B  have  the  same  income  ;  A  saves  |  of  his  ;  but  B,  by  spendin* 
30  ciollars  per  annum  more  than  A,  at  the  end  of  8  years  finds  himself  49 
dollars  in  debt  ;  what  is  their  income,  and  what  does  each  spend  per  annum  ? 
Ans,  their  i?icome  is  200  do'ls,  }ier  arm,  A  spends  1 75  dollars^  i^  B  205  fier  aim. 


Sect.  III.  II.  DISCOUNT.  S03 

Discount  is  an  allowance  made  for  the  payment  of  any  sum  of  money  be- 
fore it  becomes  due^  and  is  the  difference  between  that  sum,  due  some  time 
hence,  and  its  present  worth. 

THE/zrr*(?n/ wor^A  of  any  sum,  or  debt  due  sonae  time  hence,  is  such  a 
surn,  as,  if  put  to  interest,  would  in  that  time  and  at  tlie  rate  {ler  cent,  for 
which  the  discount  is  to  be  made,  amount  to  the  sum  or  debt,  then  due." 

RULE. 

As  the  amount  of  100  dollars,  for  the  given  lime  and  rate  is  to  100  dollars, 
so  is  the  given  sum  to  its  present  worth,  which  suUractcd  from  the  given 
$um,  leaves  the  discount. 

EXAMPLES. 

1.  What  is  the   discount  of  2.  What  is  the  present  worth  of 

E321,63due4  years  hence,  at  6  426  dollars  payable  in  4   years  and 

per  ce?ii  ?  12  days,  discounting  at  the  rate  of  5 

opERATiiON.  fier  cent. 

Dolls.  Ms.  Dolls.  25i,5l5, 
6  interest  of  \  00  cloUs,  1  ye^r. 
4  years. 

24 
100 

l?4  amount. 


Thea,  As    124  :   100  :  :   321,63 
321,«3 


124)32163,00(259,379 

321,63  given  sum. 
259,379  present  ivorth. 


Ans*,  62,251   discount. 


§  12,  (ZBtiuation  of  ^apmcnt,^. 


ij^j'.r^  -.>  y  ^  -/■  ■::»  j-^^^  < 


Equation  of  pnymcnts  is  the  finding  of  a  time  to  pay  at  oHce,  several  debts 
due  at  dilTcrcut  linRs  so  that  neither  parly  shall  sustain  loss. 

RULE. 
Multiply  each  payment  by  the  lime  at  which  it  is  due  ;  then  divide  ihd 
sum  of  the  products  by  the  sum  of  the  payments, aud  the  quotient  will  be  the 
equated  lime. 


204 


EQUATION  or  PAYMENTS. 


Sect.  III.  12. 


EXAMPLES. 


1 .  A  owes  B  1  36  dollars  to  be  paid 
in  10  months  ;  69  dollars  to  be  paid 
in  7  months  ;  and  260  to  be  paid  in  4 
months  *-  what  is  the  equated  time 
lor  the  payment  of  the  whole  ? 

OPERATION. 

136/  lOmlSeo 

96X    7=  672 

260X    4r=1040 


492  3072 

492)30  7  (6  months 
295 


120 
30 


49  2^3600(7  days. 
3444 


156 


be  paid  I 

in   10 


in 


2.  I  owe  S6!,125  to 
3  months,  -^  in  5  mor 
months,  and  the  remainder  in  14 
months  ;  at  what  time  ought  the  whole 
to  be  paid  ? 

jins.  61  months. 


3.  A  MERcHAN-r  ha*  owing  to  him 
300/.  to  be  paid  as  follows,  50/.  at  2 
months,  100/.  at  5  months  ;  and  the 
rest  at  8  months  :  and  it  is  agreed  to 
inake  one  payment  of  the  whole;  I 
demand  when  that  time  must  be  I 
jins,  6  months. 


4  A  MERCH A  If  7' owes  mc  900  dol- 
lars to  be  paid  in  96  days,  130  dollars 
in  12©  days,  500  dollars  in  80  days, 
1267  dollars  m  27  days  ;  what  is  the 
mean  lime  for  the  payment  of  the 
whole  ?  Jns.  63  days  very  nearly. 


Sect.  III.  13. 


GUAGING. 


205 


§13.  (©ua0in0. 


mmmm^ooOiVTOOOy^^^ 

Gu AGING  is  takinc^  the  dimensions  of  a  cask  in  inches  to  find  its  content 
in  gallons  by  the  loUowing 

METHOD. 

1.  Add  two  thirds  of  the  difference  between  the  head  and  bung  diameters 
to  the  head  diameter  for  the  mean  diameter  ;  but  if  the  stares  be  but  little 
curving  from  the  head  to  the  bung,  add  only  six  twnths  of  this  difference. 

2.  Square  the  mean  diameter,  which  muliiplkd  by  the  length  of  the  cask 
and  the  product  divided  by  294,  for  wine,  or  by  359  for  ale,  the  quotient  will 
be  the  answer  in  gallons. 

EXAMPLE. 

1.  How  many  ale  or  beer  gallons  will  a  cask  hold,  whose  bung  diameter  is 
31  inches,  head  diameter  35  inches,  and  whose  lengtli  is  36  inches  ? 


operation; 


31  Bung  di,am 
25  Bead  diam. 

25    Head  diam. 
4   Tnvo  thirds  diff. 

6  Difference. 

29. Mean  diam. 
29 

261 
58 

841  Square  of  mean  diami 
36  Length. 

5046 
2523 

359)30276(84  galls.  1  \\^ql3. 


Note.  1.  In  taking 
the  length  of  the  cask, 
an  allowance  must  be 
made  for  the  thick- 
ness for  both  heads  of 
1  inch,  1^  inch,  or  2 
inches  according  to 
the  fize  of  the  cask. 

NoTK.  2.  The  head  di- 
ameter must  be  taken  close 
to  the  chimes,  and  for 
small  casks,  add  3  tenths  of 
an  inch  ;  for  casks  of  40  or 
50  gallons,  4  tenths,  and  for 
larger  casks,  5  or  6  tenths, 
and  the  sum  will  be  very 
nearly  the  head  diameter 
within. 


§  u.  .H^ccfianicd  J?otocr.^. 


1.  Of  the  Lever. 


To  find  nvhal  weight  man  be  raised  or  balanced  by  ^ny  given  /lower,  Say,  as 
tlie  distance  between  the  body  to  be  raised  or  balanced.,  and  the  fulcrum  or 
prc/u  is  to  tiie  distance  between  the  prop  and  the  point  where  the  power  is 
ai)pUcd  ;  so  rs  the  power  to  the  weight  which  it  will  balance  or  raise. 


206  MECHANICAL  POWERS.  Sect.  IIL  14 


EXAMPLE. 

If  a  man  weighing  150/6.  rest  on  the  end  of  a  lever  12  feet  long  ;  what 
weiglit  will  he  balance  on  the  othpr  end,  supposing  the  prop  ij  foot  from 
the  weight  ? 

12  feet  the  Lever. 
1,5  distance  of  the  no  eight  from  the  fulcrum. 


10,5  distance  irom  the  Fulcrum  to  the  man.     Therefore, 
Ifeet.     Feet.  lb.  lb: 

M    1,5    :   10,5   :   :    150   :    1050  Jns. 

2.  Of  the  Wheel  and  Axle. 

As  the  diameter  of  the  axle  is  to  the  diameter  of  the  wheel  :  so  is  the 
power  applied  to  the  wheel,  to  the  weight   suspended  by  the  axle. 

EXAMPLES. 

\.  A  MECHANIC  wishes  to  make  a  windlass  in  such  a  manner,  as  that  Mb, 
applied  to  the  wheel,  should  be  equal  to  12  suspended  on  ll»e  axle  ;  now, 
supposing  the  axle  4  inches  diameter,  required  the  diameter  of  the  wheel  ? 

lb.     in.       lb.       in. 
As   1    :  4  :  :    12  :  48  jins.  or  diameter  of  the  wheel. 

2.  Suppose  the  diameter  of  the  axle  6  inches  and  that  of  the  wheel  6Q 
incites  ;  what  power  at  the  wheel  will  balance  \Olb.  at  the  axle  ?     Ans.   1  lb. 

3.  Oi  the  Screw. 

The  power  is  to  the  weight  to  be  raised  as  the  distance  between  two  threads 
ot  the  screw  is  to  the  circuniference  of  a  circle  described  by  the  power  ap- 
plied ai  the  end  of  the  lerer. 

NoTK.  1.  To  find  drcuinfer^nce  of  the  circle  described  by  the  end  of  t he 
lever.,  multiply  the  double  of  the  lever  by  3,14159  the  product  will  be  the  cirr 
cumference. 

Note*  2.  It  is  usual  to  abate  ^  of  the  eifect  of  the  machine  for  friction. 

EXAMPLES. 

There  is  a  screw  v,?hose  threads  are  an  inch  asunder  ;  the  lever  by  which 
it  is  turned  is  36  inrheslong,  and  the  weight  to  be  raised  a  ton,  or  22401b. 
What  power  or  force  must  be  applitd  to  the  end  of  the  lever  suflkient  to 
turn  the  screw,  that  is,  to  raise  the  weight  ? 

The  lever  36X2— 72-}-3, 14159— 226,1944-r/t<;  circumference, 
circiimf.     in  lb.  lb. 

Then,a«  226,194  :    1   :  :  2240  .  9,903 

Problems. 

1.  The  diameter  of  a  circle  being  given  to  find  the  circumference^  multiply 
the  diameter  by  3,14159  ;  the  product  will  be  the  circumference. 

2.  To  find  the  area  nf  a  circle  ^  the  diameter  being  given  ;  niulliply  the  square 
of  the  diajaieter  l)y  ,78539-8  ;  ttic  product  is  the  area. 

3.  To  measure  the  solidity  of  any  irregular  body  nvhose  dimensions  cannctbe 
taken  ;  put  the  body  into  some  regular  vessel  and  fill  it  with  water,  then  tak- 
ing out  the  body,  measure  the  fall  of  water  in  the  vessel  ;  if  the  vessel  be 
fjquare,  multiply  the  side  by  itself,  and  the  product  by  the  fall  ^f  water  which 
gives  the  solid  content  cf  the  irregular  body. 


SECTION  IV. 

Mil  Tiri'ii  :  nr  irr  --nt— 

Miscellaneous  ^icsdons. 

In  tliis  Section  there  is  nothing  new  to  be  proposed  to  the  Scholar.  Enough 
of  Arithmetic  has  been  taught  him  for  all  ordinary  occurrences  in  life.  It 
only  remains  to  lead  him  into  some  reflections  on  the  foregoing  rules.  For 
this  purpose  the  following  questions  are  subjoined.  They  are  Irft  without 
ansvjcrs^  that  the  Scholar's  only  resource  of  knowledge  for  \Torking  them 
should  be  in  his  own  mind.  Masters  having  wroughiout  these  questions  at 
a  leisure  hour,  may  transcribe  them  with  their  answers  inio  a  manuscript  for 
their  private  use,  to  which  on  any  Occasion,  without  trouble  or  hindrance, 
they  may  readily  advert  to  satisfy  the  enquiries  of  their  pupils. 

i.  The  Northern  Lights  wtre  first  observed  in  London  in  1360  ;  how 
many  years  since  ? 

2.  What  number.  Muliplied  by  43  produces  88159  ? 

3,  If  a  cannon  may  be  discharged  twice  with  6/^.  of  powder,  how  many 
limes  will  7C.  Sgrs,  \7lb.  discharge  the  «arae  piece  ? 

4  Reduce  14  guineas  and  £75  13s.  6|f/  to  Federal  Money. 

5.  What  is  the  interest  of  §79,49  one  year,  and  Rve  months  ? 

6.  A  OWED  B  S3 17,19,  for  which  he  gave  bis  note,  on  interest,  bearing 
4ate  July  12tb,  1797. 

.On  the  back  of  the  note  are  these  several  endorsements,  vir. 
Oct.  17,  1797,   Received  in  cash,  g6l,10. 

March  20th,   1798,  Received  17  Cwr.  of  Beef,   at  S  4,33  per  cwt. 
Jan.  1st,  1800,  Received  in  cash,  84  dollars. 
What  was  there  due  from  A  to  B,  of  principal  and  interest,  Sept.  18th, 
1801  ? 

7.  What  cost   13^  yards  of  flannel  at  US^  per  yard  ? 

8.  What  must  1  give  for  5  Cwr.  'igrs.  13/6.  of  cheese,  at  7  cents  per  lb  f 

9.  What  will  35  yards  of  broadcloth  cost  at  236G  yer  yard  ? 

10.  What  will  be  the  cost  of  a  line  of  Veal,wcighing  \6\lb.-A\.  2jf/.per/6.? 

11.  What  will  Q7\lb.  of  tallow  cost,  at  9\d.  per  lb  ? 

12.  What  will  196  yards  of  tape  cost,  at  3  farthings  per  yard  ? 

13.  What  will  56  bushels  of  oats  cost,  at  2*-.  3{d.  per  bushel  ? 
J 4.  At  >C3  7s.  6d.  fier  cwt.  for  sugar,  what  is  that  per  lb. 

15.  How  much  in  length  of  a  board  that  is  10  inches  wide  will  it  require 
to  make  a  square  foot  ? 

16.  How  many  square  feet  in  a  board  1  foot,  3  inches  wide,  and  14  feet,  9 

inches  lojig  ? 

17.  How  much  wood  in  a  load  9  feet  lofig,  3^  feet  wide,  and  2  feet  9  inchcB 
high  ? 

18.  At  SU33  per  yard  for  cloth,  what  must  I  give  for  72  yards  ? 

19.  If  21  civt.  of  cotton  wool  cost  C\  I   17*.  6c/.  what  is  that  per  lb  ? 

20.  If  1832^  gallons  of  wine  cost  /;44  6*.  what  is  that  per  gallon  ? 

21.  What  wili  53-i/<!>.  of  beef  cost  at  5  cents  5  n»ills  per/<i..^ 

22.  What  wili  SObu&hcJs  of  potatoes  cost  at  21  cents  per  bushel 


208  MISCELLANEOUS  QUESTIONS.       Sect.  IV. 

23.  At  SlO,76  per  civt.  for  sugar,  what  is  that  per  lb. 

24.  What  \\\\\  be  a  man's  wages  for  6  months,  at  43  cents  per  day,  work- 
ing: 5^  days  per  week  ? 

25.  What  must  I  ^ive  for  pasturing  my  horse  19  weeks,  at  33  cents  per 
week  ; 

26.  How  many  revolutions  does  the  moon  perform  in  144  years,  2  days, 
10  hours  ?  one  revolution  being  in  27  days,  7h.  43in. 

27.  What  will  7  pieces  of  cloth,  containing  27  yards  each,  come  to  at  155. 
A^^d.  per  yard  ? 

28.  A  MAN  spends  23  dollars  69  cents,  5  mills,  in  a  year,  what  is  that  per 
day  ? 

29.  Suppose  the  Legislature  of  this  state  should  grant  a  tax  of  7  cents  3 
mills  on  a  dollar,  what  will  a  man's  tax  be,  who  is  142  dollars  40  cents  on 
the  liu  ? 

30.  A  Bankrupt,  whose  effects  are  3948  dollars,  can  pay  his  creditors 
but  28  cents,  5  mills  on  a  dollar  :  wliat  does  he  owe  I 

31.  Suppose  a  cistern  having  a  pipe  that  conveys  4  gallons,  2  qts.  into  it 
in  an  !)our,  has  another  that  lets  oul  2  gallons,  1  qt.  Ipt.  in  an  hour  :  if  the 
cistern  contains  84  gallons,  in  what  lime  will  ii  be  filled  ? 

32.  If  so  dollars  worth  of  provisions  will  serve  20  men  25  days,  what 
number  of  men  will  the  same  provision  serve  10  days  ? 

33.  If  6  men  spend  16  dollars,  7  cents  in  40  days:  hov7  long  will  135 
men  bc'in  spentling  100  dollars  ? 

34.  A  BRIDGE  built  across  a  river  in  6  months,  by  45  men,  was  washed  a- 
way  by  the  current  ;  required  the  number  of  workmen  sufficient  to  build 
another  of  twice  as  much  worth  in  4  months  ?  > 

35.  Four  men,  A,  U,  C,  and  D,  found  a  purse  of  money  containing  12  dol- 
lars, they  agree  tliat  A  shall  have  one  third,  B  one  fourth,  C  one  sixth,  and 
D  one  eighth  of  it ;  what  must  each  one  have  according  to  this  agreement  ? 

36.  A  certain  usurer  lent  90/.  for  12  months,  and  received  principal 
and  interest  95/.  8*.  I  demand  at  what  rate  per  cent,  he  received  interest? 

S7.  Tf  a  gentlemen  have  an  e&tate  of  1000/.  per  ann.  how  much  may  he 
spend  per  day  to  lay  up  tiiree  score  guineas  at  the  year's  end  ? 

38.  What  is  the  length  of  a  road,  which  being  33  feet  wide  contarins  an 
acre  ? 

39.  Required  a  number  from  which  if  7  be  swbtracted  and  the  remain- 
der f>e  divided  by  8,  and  the  quotient  be  mullipiitd  by  5,  and  4  edded  lo  the 
product,  the  square  root  of  the  sum  extracted,  and  three  fourths  of  that  root 
cubed,  the  cube  divided  by  9,  the  last  quotient  may  be  24  ? 

40.  If  a  quarter  of  wheat  afford  60  ten  penny  loaves,  how  many  eight 
penny  loaves  may  be  obtained  from  it  ? 

41.  If  the  carriage  of  7  cwt.  2qr.  for  105  miles  be  1/.  5s,  how  far  may  5 
cwt.    1  qp.  be  carried  for  the  same  money  ? 

42.  If  50  men  coi.>sume  15  bushels  of  yrain  in  40  days,  how  much  will  30 
men  consume  in  60  days  ? 

43.  On  the  same  supposition,  how  long  will  50  bushels  maintain  64  men  ? 

44.  A  gentleman  having  50.9.  to  pay  among  his  laborers  for  a  days  work, 
would  give  to  every  boy  6d.  to  every  woman  8d.  and  to  every  man  16d.  the 
the  numberof  boys,  women  and  men,  was  the  same,  I  demand  the  number 
of  each  ? 

45.  A  gentleman  had  71.  \7s.  6:1.  to  pay  among  his  laborers  ;  to  every 
boy  he  gave  t>d.  to  every  w'oman  8<i.  and  to  every  man  16c/.  and  there  were 
for  every  boy  three  women,  and  for  every  woman  two  men  j  I  demand  the 
liumber  of  each  ?   ' 


1 


MISCELLANEOUS  QUESTIONS*  209 

46.  TrtREii  Gardners,  A,  B  and  C,  having  bought  a  piece  of  Ground,  find 
the  profits  ofit  amount  to  120/.  per  annunn.  Now  the  sum  of  Money  wiiicli 
they  laid  down  was  in  such  proportion,  tliat  as  often  as  A  paid  5/.  B  paki  7/. 
and  is  often  as  B  paid  4/.  C.  paid  6/.  1  demand  how  much  each  man  must 
have  per  annum  of  the  ^ain  ? 

47.  A  YOUNG  man  received  210/.  wliich  was  -1^  of  his  eldest  Brother's  por- 
tion ;  No\T  three  times  the  eldest  brother's  portion  was  half  of  the  Father's 
Estate  ;  1  demand  how  much  the  estate  was  ? 

48.  Two  men  depart  both  from  one  places  the  one  goes  North  and  the  eth- 
er South  ;  the  one  goes  7  miles  a  day,  the  other  1 1  miles  a  day  ;  how  far 
are  they  distant  the  12th  day  after  their  departure  ? 

49  Two  men  depart  both  from  one  place  and  both  ^o  the  rame  road  ;  the 
one  travels  12  miles  every  day,  the  other  17  miles  every  day  ;  how  far  are 
they  distant  the  10th  day  after  their  departure  ? 

50.  The  river  Po  is  lOOO  feet  broad,  and  10  feet  deep,  and  it  runs  at  the 
rate  of  4  miles  in  an  hour.  In  what  time  will  it  di-scharge  a  cubic  mile  of 
water  (reckoning  5000  feet  to  the  mile)  into  the  sea  ? 

51.  If  the  country  which  supplies  the  river  Po  with  water  be  f"^ 80  miles 
long  and  120  broad,  and  the  whole  land  upon  the  surface  of  the  earth  be 
62,700,000  square  miles,  and  if  the  quantity  of  water  discharej;ed  by  the  riv- 
ers into  the  sea  be  every  where  proportional  to  the  extent  of  land  by  which 
the  rivers  are  supplied  ;  how  many  times  greater  than  the  Po  will  the  whole 
amount  of  the  rivers  be  I 

52.  Upon  the  same  supposition,  what  quantity  of  water  altogether  will  be 
discharged  by  all  the  rivers  into  the  sea  in  a  year  ? 

53.  If  the  proportion  of  sea  on  the  surface  of  the  earth  to  that  of  land  be 
as  10  V  to  5  and  the  mean  dcptii  of  the  sea  be  a  quarter  of  a  mile  ;  how  many 
years  would  it  take  if  the  ocean  were  empty  to  fill  it  by  the  rivers  running  at 
the  present  rate  ? 

54.  If  SI  cubic  foot  of  water  weigh  1000  or.  avoirdupois,  and  the  weight  of 
mercury  be  13^  times  greater  than  of  water,  and  the  height  of  the  mercury 
in  the  barometer  (the  weight  of  which  is  equal  to  the  weight  of  a  column  of 
air  on  the  same  base,  extending  to  the  top  of  the  atmosphere.)  be  30  inches  ; 
what  win  be  the  weight  of  the  air  upon  a  square  foot  ?  a  square  mile  ?  and 
what  will  be  tiv^  whole  weight  of  the  atmosphere,  supposing  the  size  of  the 
earth  as  in  questions  5  1   and  53  ? 

55.  A  BEGAN  trade  June  1,  with  40  dollars,  and  took  in  B,  as  a  partner* 
Sept.  8,  following,  with  120  dollars  ;  on  Dec.  24,  A  i)ut  in  190  dollais  morei 
and  continued  the  whole  in  trade  till  May  5,  following,  when  their  whole  gain 
was  found  to  be  82  dollars  ;  what  is  each  partnen's  share  ? 

56.  If  I  give  80  bushels  of  potatoes,  at  21  cents  per  bushel,  and  240//*  of 
flax,  at  15  cents  per  lb.  for  64  bushels  of  salt,  w!iat  is  the  *alt  per  bushel  ? 

57.  What  is  the  present  worth  of  482  dollars  payable  4  years  hence,  dis- 
counting at  the  rate  of  6  per  cent  ? 

58.  1  have  owing  to  me  as  follows  viz.  §18,73  in  8  months  ;  S46,00  in 
5  months  ;  and  S1C)4,84  in  three  months  ;  what  is  the  meantime  for  the 
payment  of  the  whole  ? 

59.  If  I  sell  500  deals  at  15c/.  a  piece,  and  lose  C9  per  cent,  what  do  I 
lose  in    the  whole  quantity  ? 

60.  If  I  buy  1000  Ells  Flemish  of  linerj  for  /i'^0  what  may  I  sell  it  ;^cr 
Ell  in  London  to  gain  yClO,  iu  the  whole  ? 


210  MISCELLANEOUS  QUE:sTIONS. 

Pleasant  and  diiicrting  ^icstzons. 

1.  TwEiiE  was  a  well  30  fv.^ct  deep  ;  a  p' rot;;  at  ihe  bottom  could  jump  tip 
3  feet  every  clay  but  he  would  fall  bark  two  feet  every  right.  How  many 
days  did  it  take  the  Frog  to  jump  out  ? 

2.  Two  men  were  drivir>ij  sheep  to  market,  says  one  to  the  other,  give  me 
one  of  yours  and  I  shall  have  au  many  as  yovi  ;  tiie  otiier  says,  give  me  on«  of 
yours  and  I  shall  have  as  many  again  i-.s  you.     How  many  hud  each  ? 

3.  As  I  was  goin^to  St.  IveSi 
I  flnet  seven  wives, 
Every  wife  had  seven  sacks. 
Every  sack  had  seven  cuts, 
Every  cat  had  seven  kits, 
Kits,  cats,  sacks,  and  vvives, 
How  many  vvere  going  to  St.  Ives  ? 

4.  The  account  of  a  certain  School  ir,  as  follows,-  \(iz.  y^^  of  the  boys  learn  . 
geometry,!  learn  grammar,  J'j  learn  arithmetic,  ^^^  learn  to  write  and  9  learn 
to  read  ;  1  demand  tke  number  of  each  ? 

5.  A  MAN  driving  his  geese  to  market^  was  met  by  another,  who  said,  Good-  - 
morrow,  master,  with  your  hundred  geese;  snys  he  1  have  not  an  hundred, 
but  if  I  had  half  as  many  as  I  now    have,  and  two  geese  and  a  half  beside  the 
number  I  now  have  already,  I  should  have  an  hundred  :  How  many  had  he  .••  , 

6.  Three  travellers  met  at  a  caravansary,  or  iim,  in  Persia  ;  and  two  of 
them  brought  their  provisions  along  with  them,  accoj^g  to  the  custom  of 
the  country  }  but  the  third  not  having  provided  aiijfpropostd  to  the  others 
that  they  should  eat  together,  and  he  would  pay  tl«?alue  of  his  proportion. 
This  being  agreed  to,  A  produces  5  loaves,  and  B  vSlbaves,  which  the  travel- 
lers eat  together,  and  C  ptiid  8  pieces  of  money  as  the  vahfe  of  his  share,  with  ' 
which  the  others  were  aaiisfred,  but  quarreled  about  the  dividing  of  it.  Up- 
on this  the  affair  was  referred  to  the  judge  who  decided  the  dispute  by  an 
impartial  sentence.     Required  his  decision  ? 

7.  Suppose  the  9  Pigits  to  be  placed  in  a  quadrangular  form  :  I  demand- 
in  what  order  they  must  standi  that  any  three  figures  in   a  right  line,  may 
make  just   15. 

8.  A  Countryman  having  a  FoK,  a  Goose,  and  a  peck  of  Corn,  in  his  jour- 
ney came  to  a  river,  where  it  so  happened  that  he  could  carry  but  one  over  at 
a  time.  Now  as  no  two  were  to  be  left  together  that  might  destroy  each, 
other;  so  he  was  at  his  M'its  end  how  to  dispose  of  them  :  for,  Suys  he, 
the  corn  can't  eat  the  goose,  nor  the  goose  eat  the  fox,  yet  the  fox  can  eat  the 
goose,  and  the  goose  eat  the  corn.  The  Question  is  how  he  must  carry  them 
over,  that  the),'  miglit  nut  devour  each  other- 

9.  Three  jealous  husbands  with  their  wives,  being  ready  to  pass  by  night 
over  a  river,  do  fmd  at  water  side  a  boat  which  can  carry  but  two  persons  at, 
once,  and  for  want  of  a  waterman  they  arc  necessitated  to  row  themselves  over 
the  river  at  several  times  :  The  Question  is,  how  those  6  persons  shall  pass 
by  2  and  2,  so  that  none  of  the  three  wives  may  be  fouiicl  in  the  company  of 
one  or  two  men,  unless  her  husband  be  present  ? 

10.  Two  merry  companioris  are  to  have  equal  shares  of  8  gallons  of  wine, 
which  are  in  a  vessel  containing  exactly  8  s^allons  ;  now  to  divide  it  equally 
betweew  them  they  have  only  tv/o  oth.er  empty  vessels,  of  wldch  one  contains 
5  gallons,  and  the  other  3  ;  Thequesiion  is,  how  they  shall  divide  the  said 
wine  between  them  b/thc  help  of  these  three  vessels,  so  that,  they  may  hijive 
4  gallons  fi-piece  ?  ^ 

/" 


SECTION  III. 

Forms  oj  Notes  J  Deeds,  Bomls,  and  other  Instruments  of  IFriting^ 

§  1.  Of  Notes. 

No.  I.  , 

Overdcan^  Scfit.  17,  1  802.     For  value  received,  T  promise  to  pay  to  Oliver 
Bounliful,  ©r  order,  Sixty -three  dollars  fifty -four  cents  on  demand,  with  in- 
terest after  three  months,  William  Triisty, 
Attest,  Timothij   Testimony. 

NO.  II. 

Bilforli  Se/it.  17,  18©2.     For  value  received,  I  promise  to  pay  to  f).  R.  04j 

bearer       >.    .dollars    cents,  three  months  after  date. 

Feter  Pencil. 


No.  III. 


V 


Br   riVO    PERSONS. 


Arian,  Sefit.  17,  1502.     For  value,  received  we  jointly  and  severally  prom* 

ise  to  pay  to  C.  D.  o*r  order— dollars cents  on  demand,  with  interest. 

Attest,  Mden  Faithfuls 

Constance  jidley^  James  Fairface. 

'  OBSERVATIONS. 

1.  No  note  is  negotiable  unless  the  words,  or  order ^  otherwise,  or  bearer^ 
be  inserted  in  it, 

2.  If  the  note  be  written  to  pay  him  '•  or  order,**  {No.  I.)  then  Oliver  Bonn' 
tiful  may  endorse  this  note,  tliatis,  write  his  name  on  the  back  side,  and  sell 
it  to  A,  B,  C,  or  whom  he  pleases.  Then  A,  who  buys  the  note,  calls  on  WU^ 
Ham  Trusty  for  payment,  and  if  he  neglects  or  is  unable  to  pay,  A  may  re- 
cover it  of  the  Endorser. 

3.  If  a  note  be  wiitten,  to  pay  him  "  or  bearer,**  (No.  II.)  then  any  persoa 
who  holds  the  note  may  sue  and  recover  the  same  of  Peter  Pencil. 

4.  The  rate  of  interest  established  by  law  being  sijc  fier  cent.fier  annum,  it 
becomes  unnecessary,  in  writing  notes  to  mention  the  rate  of  interest  ;  it  is 
suffiicient  to  write  them  for  the  payment  of  such  a  sum,  with  interest,  for  it 
will  be  underBlood,  legal  interest,  which  is  .six  per  cent. 

5.  All  notes  are  either  payable  on  demand,  or  at  the  expiration  of  a  cer- 
tain term, of  time  agreed  upon  by  the  parties  and  mentioned  in  ths  note,  ai 
three  months,  or  a  year,  £cc. 

6.  If  a  bond  or  note  mention  no  time  of  payment,  it  is  ulwaya  oademand^ 
whether  the  words  *'  on  demand"  be  expressed  or  not. 


212  FORMS  OF  BONDS. 

7.  All  notes  payable  at  a  certain  time  are  on  interest  as  soon  as  they  be- 
come clue,  ihough  in  such  notes  there  be  no  mention  made  of  interest. 

This  rule  is  founded  on  the  principle,  that  every  man  ought  to  receive  his 
money  when  due,  and  that  the  nonpayment  of  it  at  that  time  is  an  injury  to 
him-  The  law,  therefore,  lo  do  him  justice,  allows  him  interest  from  the  time 
the  money  becomes  due,  as  a  compensation  for  the  injury. 

8.  Upon  t'le  same  principle  a  note  payable  on  demand,  witkout  any  men- 
tion made  of  interest  is  on  interest  after  a  demand  of  paymeJit,  for  upon  de- 
mand, such  notes  immediately  become  due. 

9.  If  a  note  be  given  for  a  specifie  article,  ns  rye  payable  in  one,  two  or 
tliree  months,  or  in  any  certaiis  time,  and  the  signer  of  such  note  suffers  the 
time  to  elapse  without  delivering  such  ariicle,  the  holder  of  the  note  will  not 
be  oblij^ed  to  take  the  article  afterwards,  but  may  demand  and  recover  the 
value  of  ii  in  money. 

§  2.  Of  Bonds. 

A  Bond,  iviib  a  condition  J rom  one  to  another. 

Know  all  pnen  by  these  presents,  that  I  C.  D.  of,  Sec.  in  the  county  of  Sec. 
am  held  and  firmly  bound  to  E  F.  of,  Sec.  in  two  hundred  dollars  to  be  paid 
lo  the  said  E.  V ■  or  his  certain  attorney,  his  executors,  administrators  or  as- 
sip;ns  ;  to  which  payment,  well  and  truly  to  be  made,  I  bind  myself,  my 
heirs,  executors,  and  administrators,  firmly  by  these  presents  ;  Sealed  with 
my  seal.  Dated  the  eleventh  day  of in  the  year  of  our  Lord  one  thous- 
and eight  hundred  and  two. 

The  condition  of  this  oblig-ntion  is  such,  That  if  the  above  bound  C.  D.  his 
heirs,  executors  or  administrators,  do  and  shall  well  and  truly  pay  or  cause 
to  be  paid,  unto  the  above  named  E.  F.  his  executors,  administrators  or  as- 
signs, the  full  sumof  two  hundred  dollars,  with  legal  interest  for  the  same,  on 

or  before  the  eleventh  day  of--; next  ensuing  the  date  hereof:   Then  this 

obligation  to  be  void,  or  otherwise  to  remain  in  full  force  and  virtue* 

A  Condition  of  a  Counter  Bond,  or  Bond  of  Indemnity,  where  one 
man  becomes  bound  for  another. 

The  condition  of  this  obligation  is  such,  That  whereas  the  above  named 
A.  B  at  the  special  instance  and  request,  and  for  the  only  proper  debt  of  the 
above  bound  C.  D.  together  with  the  said  C.  D.  is,  in  and  by  one  bond  or  ob- 
ligation bearing  equal  date  with  the  obligation  above  written,  held  and  firmly 
bound  unto  E.  F.  of  Sec.  in  the  penal  sum  of dollars,  condi- 
tioned for  the  payment  of  the  sum  of,  Sec.  with  legal  interest  for  the  same  on 

the r-day  of next  ensuing  the  date  of  the  said  in  part  recited  obligation, 

as  in  and  by  the  said  in  part  recited  bond,  with  the  condition  thereunder  writ- 
ten may  more  fully  appear  :  If  therefore  the  s«id  C.  D.  his  heirs,  executors 
or  administrators,  do  and  shall  well  and  truly  pay  or  cause  to  be  paid  tinto 
the  said  E.  F.  his  executors,  administrators  or  assigns,   the  said  sum  of  Sec. 

with  legal  ii-terest  for  the  same,  on  the  said day  of,  Sec.  next  ensuing  the 

dale  of  the  said  in  part  recited  obligation,  according  to  tiie  true  intent  and 
meaning,  and  in  full  discharge  and  satisfaction  of  the  snid  in  part  recited  bor.d 
or  oblig'iaion  :  Then,  Sec.  Otherwise,  Sec- 


FORMS  OF  RFXEIPTS,  2ia 

A^oTe.  The  principal  difference  between  a  note  and  a  bond  is  that  the  lat^ 
ter  is  an  instrument  of  more  solemnity,  being  given  under  seal.  Also,  a  note 
may  be  controuled  by  a  special  ag-ceement,  different  from  the  note,  where- 
as, in  case  of  a  bond,  no  special  agreement  can  in  the  least  controul  what 
appears  to  have  been  the  intention  of  the  parties  as  expressed  by  the  word* 
in  the  condition  of  the  bond. 

§.  3.  Of  Receipts, 

No.  I. 

Sifgrieves,  Se/tt.  19,  1802.  Received  from  Mr.  Durance  Adley,  tea 
dollars  in  full  of  all  accounts. 

Orvand  Constance^ 

No.  II. 

Sitgrieves'i  Scfit.  19,  1802.  Received  from  Mr.  Orvand  Constance,  five  dol- 
lars in  full  of  all  accounts. 


Durance  Adley ^ 


No.  III. 


A  receipt  for  an  endorsement  on  a  Note 


Siigrieves,  Sefit.  19,  1802.  Received  from  Mr.  Simpson  Easily,  (by  the 
Iiand  of  Mr.  Titus  Trusty),  sixteen  dollars  twenty  five  cents,  which  is  en- 
dorsed on  his  note  of  June  3,  1802. 

Feter  Cheerful. 

No.  IV. 

A  receipt  for  money  received  on  Account. 

Stf^^rxeves,  Sept,  19, 'l 802.  i^cceived  ©f  Mr.  Grand  Landlike,  fifty  dollars 
on  account. 

^  Eldro  Slackley. 


No.  V. 


A  Receipt  for  interest  due  on  a  Bond. 


Received  tliis day  of of  INIr.  A.  B.  the  sum  of  fire  pounds,  in 

full  of  one  year's  interest  of  100/.  due  to  me  on  the day  of last, 

Qn  bond  from  the  said  A.  B.  1  say  received-  By  me  C.  D. 


Observations. , 

1 .  Therk  is  a  distinction  between  receipts  given  in  full  of  a//  accounts^  and 
otiicrs  in  full  oi  all  devuinds.  The  former  cut  off  accounts  only  ;  the  latter 
cut  of  not  only  accounts,  but  all  obligations  and  riglit  of  action. 

2.  Whkm  any  two  persons  make  a  settlement  and  pass  receipts  (No.  Land 
No.  II  )  each  receipt  must  specify  a  particular  sum,  rectivcd,  less  or  more. 
It  is  not  necessary  that  the  sum  specified  in  the  receipt,  be  the  exact  sum  re- 
ceived. 


$14  FORMS  OF  ORDERS. 

§  4.   Of  Orders. 

No.  I. 

Mr.  Stefihen  Burgess. 
SIR, 
For  value  received,  pay  to  A.  B.  Ten  Dollars,  and  place  the  same  to  my 
account.  Samuel  Skmner. 

Archdale,  Sept.  9th,   1802. 

No.  II. 

Boston,  Se/it.9t/i,  1802. 
SIR, 
For  value  received,  pay  G.  R.  eighty  sij^  cents,  and  this,  with  his  receipt, 
shall  be  your  discharge  from  rne. 

Nicholas  Reuhem. 
To  Mr.  James  Robottom, 

§    5.    Of  Deeds, 

Mo.  I. 

A  Warrantee  Deed. 

Know  all  mkn  by  these  presents,  That  I  Peter  Careful  of  Leominster, 
in  the  county  of  Worcester  and  Commonwealth  of  Massachusetts,  gentleman, 
for  and  In  consideration  of  one  hundred  fifty  dollars,  and  forty-five  cents  paid 
to  me  by  Samuel  Pendleton  of  Ashby,  in  the  county  of  Middlesex,  and  Com- 
monwealth of  Massachusetts,  yeoman,  the  receipt  whereof  I  do  hereby  ac- 
knowledge, do  hereby  give,  grant,  sell,  and  convey  to  the  said  Samuel  Pen- 
dleton, his  heira  and  assigns,  a  certain  tract  and  parcel  of  land,  bounded  as 
follows,  viz. 

l^Here  insert  the  boundfi,  tog^ether  ivitb  a,ll  the  privileg^es  cind  afifiurtenences  there- 
unto belonging.'l 

To  have  and  to  hold  the  same  unto  the  said  Samuel  Pendleton  his  heirs  and 
assigns  to  his  and  their  use  andfoehoof  forever.  And  I  do  covenant  with  the 
said  Peter  Pendleton  his  heirs  and  assigns,  That  I  am  lawfully  seized  in  fee 
of  the  premises,  that  they  are  free  of  all  incumbrances,  and  that  I  will  war- 
rant and  defend  the  sitme  to  the  said  Peter  Pendleton  his  heirs  and  assigns 
forever,  against  the  lawful  claims  and  demands  of  all  persons. 

In  witness  wh-sreof  I  have  hereunto  set  my  hand  and  seal,  this day 

of in  the  year  of  our  Lord  OnQ  thousand  eight  hundred  and  two. 

Signed,  sealed  and  de-  >  ^^^^^ 

hveredinfiresenccef     y  j         \^ 

L.  R. 
F.  G. 

No.  IL 

^litdaim  Deed, 

Know  all  meic  by  these  presents,  That  I  A.  B.  of,  Sec.  in  considera- 
tion of  the  sum  ot to  be  paid  by  C.  D.  of.  Sec.   the  receipt  whereof  I  do 

hereby  acknowledge,  have  remissed,  released,  and  forever  quit-claimed,  and 
do  by  these  presents  remiss  and  release,  and  forever  quit-claim  unto  the  said 
'C  D.  his  heirs  and  assigns  forever.  [Here  insert  the  firemises."]  To  have  and  to 
hold  the  same,  togethen  with  all  the  privileges  and  appertenances  thereunto 
belongingjto  him  the  said  CD.  his  heirs  and  assigns  forever. — In  witness,  life. 


FORM  OF  DEEDS,  215 

No.  III.  A  Mortgage  Deed, 

Know  all   men   by  these  presents,  That  I  Simpson  Easly,  of in 

the  county  of in  the  state  of >  Blacksmith,  in  consideration 

of Dollars Cents  paid  by  EWin  Fairface  of in  the  conn-  • 

ty  of • in  the  state  of— —Shoemaker,  the  Receipt  whereof  I   do 

hereby  acknowledge,  do  hereby  give,  grant,  sell  and  convey  unto  the  said  El- 
vin  P'airfiice,  his  heirs  and  assigns,  a  certain  tract  and  parcel  of  land,  bounded 
as  follows,  viz.  [^Here  insert  the  bounds  together  with  ail  the  privileges  and  afi- 
Jmrtenances  thereunto  belonging.']  To  have  and  to  hold  the  afore-granted  Prem- 
ises to  the  said  Elvin  Fairface,  his  heirs  and  assigns,  to  his  and  their  use  and  • 
behoof  forever.  And  I  do  ctivenant  with  the  said  Elvin  Fairface  his  heirs  and 
assigns,  That  I  am  lawfully  seized  in  fee  of  the  afore-granted  premises  ; 
That  they  are  free  of  all  incumbrances  :  That  I  -have  good  right  to  sell  and 
convey  the  same  to  the  said  Elvin  Fairface.  And  that  I  will  warrant  and  de- 
fend the  same  premises  to  the  said  Elvin,  his  heirs  and  assigns  forever,  against 
the  lawful  claims  and  demands  of  all  persons.  Frovided  nevertheless^  That  if  I 
the  said  Simpson  Easly  my  heirs,  executors,  or  administrators,  shall  well  and 
truly  pay  to  the  said  Elvin  Fairface,    his  heirs,  executors,  administrators  or 

assigns,  the  full  and  just  sum  of         dollars cents  on  or  before  the  — — 

day  of ^which  will  be  in  the  year  of  our  Lord  Eighteen  Hundred  and — — - 

with  lawful  interest  tor  the  same  until  paid,  then  this  deed,  as  also  a  certain 
bond  [or  note  as  the  case  may  6e] bearing  date  with  these  presents  given  by  me  • 
to  the  said  Fairface,  conditioned  to  pay  the  same  sum  and  interest  at  the  time 
aforesaid  shall  be  void  ;  otherwise  to  remain  in  full  force  and  virtue.  In  wit- 
Rpss  whereof,  I  the  said  Simpson  and  Abigail  my  wife,  in  testimcney  that  she 
Telinquishes  all  her  right  to  dower  or  alimony  in  and  to  the  above  described. 

preiTiises,  have  hereunto  set  our  hands  and  seals  this — —sna^i — day  of — 

in  the  year  of  our  Lord  one  thoiuand  eigJu  hundred  and  five. 

Signed,  sealed^  and  de-'^  Simfison  Easly .    Q 

livercdinfirescjice  of.        3  ^bigaii  £a»ly,      q. 

L.  N. 
V.  X. 

§  6.  Of  an  Indenture. 

J  common  Indenture  to  bmd  an  Apprentice, 

This  Indenture  Witnesneth.  That  A.  B.  of,  &c.  hath  put  and  placed,  and  by 
these  presents  doth  put  and  bind  out  his  son  C.  D.  and  the  said  C.  D.  doth 
hereby  put,  place  and  bind  out  himself,  as  an  apprentice  to  R.  P.  to  learn  the 
art,  trade  or  mistery  of The  said  C.  D.  after  the  manner  of  an  appren- 
tice, to  dwell  with  and  serve  the  said  R.  P. from  the  day  of  the  date  here- 
of, until  the day  of which  will  be  in  the  year  of  our  Lord  one  thousand. 

eight  hundred  and at  which  lime  the  said  apprentice  if  he  should  be  liv- 
ing, ^vill  bo  twenty-one  years  of  age  :  During  which  time  or  term  the  said 
apprentice  his  said  master  well  and  faithfully  shall  serve  ;  his  secrets  keep, 
and  his  lawful  coiiimands  every  where  at  all  times  readily  obey  ,  he  shall  do 
no  damage  to  his  said  master,  nor  wilfully  sufler  any  to  be  doneb>^oth- 
ers  ;  and  if  any  to  his  knowledgo  be  intended,  he  shall  give  his  master  season- 
able notice  thereof.  He  shall  not  waste  the  goods  of  his  said  muster  nor 
lend  them  unlawfully  to  any  ;  at  cards,  dice,  or  any  other  unlawful  game  he 
shall  not  play  ;  fornication  he  shall  not  commit,  nor  matrimony  contract  dur- 
ing the  said  term  ;  taverns,  ale-houses,  or  places  of  gaming  he  shall  not  haunt 
or  frequent  :  From  the  service  of  his  said  master,  he  shall  not  abjent  him 
self;  butia  all  things  and  at  all  limes  he  shall  carry  and  behave  himself  as  n 
good  and  faithful  apprenlice  ought,  during  the  whole  lituc  or  term  aforesaid 


§ 


^IG  iFORiM  OF  A  WILL* 

And  the  said  R.P.  on  l.is  part,  cloth  hereby  promise^  covenant  and  agree  to 
teach  and  instruct  the  said  apprentice,  or  cause  him  to  be  taught  and  instruct- 
ed, in  the  art>  trade  or  calling  of  a ^ ^by  the  best  way  or  means  he  can, 

and  also  to  teach  and  instruct  the  said  apprentice,  or  cause  him  to  be  taught 
and  instructed,  to  read  and  write,  and  cypher  as  far  as  the  Rule  of  'J^hret?,  if 
the  said  apprentice  be  capable  to  learn,  and  shall  w'ell  and  faithfully  find  and 
provide  for  the  said  apprentice,  good  and  sufficient  meat,  drink,  clotiiing^ 
lodging  and  other  necessaries  fit  and  convenient  for  such  an  apprentice,  dur- 
ing the  term  aforesaid,  and  at  the  expiration  thereof,  shall  give  unto  the  said 
apprentice,  two  suits  of  wearing  apparel,  one  suitable  for  the  Lord's  day,  and 
the  other  lor  working  days. 

In  testimony  whereof,  the  said  parties  have  hereunto  interchangeably  set 
their  hands  and  seals,  the -day  of in  the  year  of  our  Lord  one  thou- 
sand eight  hundred  and ^  (Seal) 

Signed.,  sealed^  and  de-  >  (Seal) 

iivercd  in  firesence  o/^  ,i  (Seal) 

7.  Of  a  WilL 

The  form  of  a  P^ill,  ivitb  a  dcoisc  oj  a  Real  Estate,  Leasehold,  CsV. 

In  the  name  oJ  GOD  Amen^  I  A,  B,  of,  Scc.bting  weak  in  body,  but  of  sound 
and  perfect  mind  and  memory  {ovyountay  say  thus^  considering  the  uncer- 
tainty of  this  mortal  life,  and  being  of  sound,  Sec.)  blessed  be  Almighty 
God  for  the  same,  do  make  and  publish  this  my  last  Will  and  Testament,  in 
manner  and  form  following  (that  is  to  say)  First,  I  give  and  bequeath  unto  my 

beloved  wife  J.  B.  the  sum  of -I  do  also  give  and  bequeath  unto  ray  eldest 

son,  G.  B.  the  sum  of 1  do  also  give  and  bequeath  uwto  my  two  younger 

sons,  J.  B.  and  F,  B.  the  sum  of apiece.     I  also  give  and  bequeath  to 

my  daughter-ini^law,  S.  H.  single  woman,  the  sum  of- which  said  several 

legacies  or  sums  of  money  I  will  and  order  shall  be  paid  to  the  said  respec- 
tive legatees,  within  six  months  after  my  decease.— I  further  give  and  de- 
vise to  my  said  eldest  son  G.B.  his  heirs  and  assigns,  ^/Mhat  my  messuage  or 
tenement,  situate,  lying,  and  being  in,  8cc.  together  with  all  my  other  free- 
hold estate  whatsoever,  to  hold  to  him  the  said  G.  B.  his  heirs  and  assigns 
forever.  And  I  hereby  give  and  bequeath  to  my  said  younger  sons  J.  B.  and 
F.  B.  all  my  leasehold  estate  of  and  in  all  those  messuages  or  tenements,  with 
the  appurtenances,  situate.  Sec,  equally  to  be  divided  between  them.  And  las' 
ly,  as  to  all  the  rest,  residue  and  remainder  of  my  personal  estate,  goods  ajia 
chattels,  of  what  kind  and  nature  soever,  I  give  and  bequeath  the  same  to 
my  said  beloved,  wife  J.  B.  whom  I  hereby  appoint  sole  executrix  of  this  my 
last  will  and  testament  ;  and  hereby  revoking  all  former  wills  by  me  made. 

In  nvilness  nvherenf  I  have  hereunto  set  my  hand  and  seal,  the day  of 

in  the  year  of  our  Lord. -— 

Signed,  scaled,  published  and  declared  by  the  above  A.  B.  (Seal) 

named  A.  B.  to  be  his  last  will  and  testament,  in 
the  pre^ence  of  us,  who  have  hereunto  subscribed 
our  namcsi  as  witnesses,  in  the  presence  of  the  tes- 
tator. ^^'  s. 

W.  T. 


T,  W. 


FINIS. 


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